diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzasbv" "b/data_all_eng_slimpj/shuffled/split2/finalzzasbv" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzasbv" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nDespite the recent popularity of Quality-Diversity (QD) algorithms \\cite{pugh2016quality, cully2017quality}, these algorithms have been limited to domains in which evaluations can be performed in simulation. This is because QD algorithms need to perform evaluations in the order of millions and where the outcomes are not safety critical or dangerous. Examples of these application domains include robotics \\cite{cully2013behavioral, cully2015robots, chatzilygeroudis2018reset}, video games \\cite{gravina2019procedural, fontaine2020illuminating} and aerodynamics \\cite{gaier2018data}. In the field of robotics, physics simulators \\cite{coumans2020, Lee2018, todorov2012mujoco} are commonly used and QD algorithms depend heavily on these to obtain abundant amounts of data and evaluations to learn behavioural repertoires of robots. However, building fast and accurate physics simulators to model the complex dynamics of robots and the wide variety of potential environments is difficult. Furthermore, even with extensive modelling of different scenarios, there is still the difficult problem of sim-to-real transfer \\cite{zhao2020sim, akkaya2019solving, lee2020learning}. \nTo realize the potential of learning and QD algorithms for robotics and to have the real-world impact we want them to have, we need algorithms which can effectively learn repertoires of skills autonomously and adapt directly in the real-world. \n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/fig1-rfqd_updated_v2.pdf}\n \\caption{Safe reset-free movement through diversity of behavioural space demonstrated in a real-world environment.}\n \\label{fig:illustration}\n \\vspace{-6mm}\n\\end{figure}\n\nIn this work, as a step towards using QD algorithms in the real-world, we address two key issues that arise when attempting to learn behavioural repertoires in the real-world in an autonomous manner: \\textit{resets} and \\textit{safety}. \nWe then aim to maximise the sample efficiency of learning behavioural repertoires while considering these constraints. We specifically focus on reset-free learning of robotic locomotion skills to highlight these issues and our approach to solving them. \n\nAn often overlooked requirement of QD algorithms when used for Reinforcement Learning (RL) is the episodic setting they function in. This requires the environment to be set to a fixed initial state at the start of every episodic trial as the behavioural descriptor is measured as a function of the trajectory of states from this initial state to the final state in the episode. In real-world settings, this would correspond to humans manually resetting the robot and environment after every episode. This is an impractical and expensive solution considering the number of evaluations that conventional QD algorithms require are in the order of millions. With the considerable amount of human supervision and instrumentation required for resets, this defeats the purpose of QD algorithms to autonomously learn complex skills. \n\nAnother key challenge of learning in the real-world is safety. Actions taken must not be dangerous to the robot and the environment. For learning locomotion skills, this corresponds to avoiding collisions with objects in the environment during learning. Achieving this while performing QD would require both the capability to predict the outcome of the execution of a new behaviour as well as an understanding of its implications with respect to its safety.\n\nFinally, we want to learn skills efficiently with efficiency being measured by the number of evaluations taken. In the real-world, this also directly corresponds to the learning time needed. The goal is to intelligently select behaviours that would help improve the behavioural repertoire while minimizing unnecessary non-informative trials.\n\nWe introduce Reset-Free Quality Diversity (RF-QD) as a framework for the real-world execution of QD algorithms (see Figure~\\ref{fig:illustration}). In a nutshell, RF-QD is a Dynamics-Aware Quality-Diversity (DA-QD) algorithm combined with a Behaviour Selection Policy to select only safe and valuable behaviours for evaluation in the (potentially dangerous) real-world. \nWe demonstrate an algorithm which autonomously acquires a diverse repertoire of locomotion skills on a hexapod robot in safety-constrained environments.\n\n\n\\section{Related Work}\n\n\\subsection{Quality-Diversity and Behavioural Repertoire Learning in Robotics} \\label{subsec:rel-QD}\nQuality-Diversity (QD) optimization is a class of algorithms that aims to generate a collection of both diverse and high-performing solutions \\cite{pugh2016quality, cully2017quality}. In the context of robotics, each solution can for instance, be a parametric policy which determines sequences of actions to execute (i.e. motor commands), resulting in a behaviour. A behaviour can then be represented by a numerical vector referred to as the \\textit{Behavioural Descriptor (BD)}. The BD is a low-dimensional representation of the trajectory of states the policy visited and is usually defined manually depending on the tasks. However, the BD can also be learned in an unsupervised manner \\cite{cully2019autonomous, paolo2020unsupervised, grillotti2021unsupervised}. The choice of the BD is important as it determines the novelty of a solution and will be used to drive the exploration to cover the BD space \\cite{lehman2011abandoning}. \n\nConceptually, QD extends the single novelty-seeking objective introduced in the Novelty Search algorithm \\cite{lehman2011abandoning} with another measure of quality. MAP-Elites \\cite{mouret2015illuminating} and Novelty Search with Local Competition \\cite{lehman2011evolving} are two commonly used and well-known QD algorithms. Cully and Demiris \\cite{cully2017quality} suggested that most QD algorithms can be represented using a common framework consisting of two key components; the \\textit{archive} and \\textit{selector}. Variants of QD algorithms develop around these components, all building on the QD loop of selection, variation, evaluation and (tentative) addition to the archive. The archive is used to store the highest performing solutions for each niche. Instead of a uniform grid used to discretize the BD space, methods like CVT-MAP Elites \\cite{vassiliades2017using} or Sliding-Boundary MAP-Elites \\cite{fontaine2019mapping} modify this to make the archive more flexible. More recently, there has also been a body of work focused on using more complex selectors and efficient optimizers such as evolutionary strategies \\cite{colas2020scaling, fontaine2020covariance} and policy gradients \\cite{nilsson2021policy, pierrot2021diversity}.\n\nIn the field of robotics, it was shown that the final archive of solutions discovered by QD algorithms can be used as a behavioural repertoire to perform downstream tasks \\cite{cully2013behavioral}. For example, each solution is a controller which is represented by a set of parametric functions which controls the robot's joints. Coupled with Bayesian optimization, the controllers generated in the archive can be used to quickly adapt to unforeseen mechanical damage, help with sim-to-real transfer and solve longer-horizon tasks using planning algorithms \\cite{cully2015robots, chatzilygeroudis2018reset}.\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.85\\textwidth]{figures\/rf-qd_figure2.pdf}\n \\caption{RF-QD performs QD in imagination (as in DA-QD) and uses a more intelligent behaviour selection policy to keep the robot in the safe regions of its environment while maximising the value gained by every real-world evaluation.}\n \\label{fig:overview}\n \\vspace{-4mm}\n\\end{figure*}\n\\subsection{Model-based Quality-Diversity} \\label{subsec:rel-MBQD}\nOne of the main bottlenecks of Quality-Diversity (QD) algorithms is the sample efficiency. QD algorithms typically require on the order of millions of evaluations and rely on parallel computation of these evaluations. This single factor alone usually make them unsuitable to be used directly in the real-world.\n\nA line of work that attempts to address this problem, now referred to as \\textit{model-based quality-diversity} methods, is through the use of models. Surrogate-Assisted Illumination (SAIL) \\cite{gaier2018data} first introduced the use of surrogate models for QD algorithms. SAIL integrates surrogate models, in the form of Gaussian Process (GP) models, to approximate the objective function and reduce the number of evaluations for the computationally expensive application of aerodynamic design. Another algorithm called M-QD \\cite{keller2020model} later follows up on this idea and used neural network models that map the parameter space to the behaviour and fitness space as a surrogate model. They demonstrate this on robotic pushing and placing tasks.\n\nDynamics-Aware Quality-Diversity (DA-QD) \\cite{lim2021dynamics} is another approach which instead uses learnt dynamics models as a surrogate model. DA-QD introduced the concept of the imagined repertoire which allows QD to be performed fully in imagination using the learnt dynamics models. The dynamics models are trained incrementally and online as data is collected through evaluations. DA-QD showed a significant ($\\approx$20 time) increase in sample-efficiency. Our work uses the DA-QD framework to heavily reduce the number of evaluations needed making it feasible to be considered for a real-world application. With RF-QD, we extend DA-QD to make better use of the imagined repertoire to select behaviours more intelligently and in a sequential manner. While DA-QD only uses variation operators for optimization in imagination, we also further study the effect of optimizing for different objectives in imagination.\n\n\\subsection{Reset-free Learning} \\label{subsec:rel-resetfree}\nReset-free Learning has mainly been studied in gradient-based Deep Reinforcement Learning (DRL) where the episodic setting is also usually a prerequisite of the Markov Decision Process (MDP) formulation of the problem. One approach taken to enable real-world RL is to automate resets using other manually scripted robots to reset objects and the environment to the initial state distribution required \\cite{nagabandi2020deep}. While this works well for resetting manipulation tasks in which the workspace is relatively limited, this approach is difficult to apply for learning locomotion behaviours.\n\nMost similar to our work and another promising method is to use a multi-task RL approach \\cite{ha2020learning, gupta2021reset}. The key idea behind this approach is to use a scheduler and the different tasks present in the multi-task setup as resets for each other. Ha et al. \\cite{ha2020learning} showed this in the context of learning simple locomotion policies while Gupta et al. \\cite{gupta2021reset} demonstrated this approach on more extensive multi-task setting to learn dexterous manipulation policies. Both these works explicitly learn policies for tasks in a pre-defined distribution of tasks. Each policy is optimized individually using an off-the-shelf deep RL algorithm and separate instances of networks and replay buffers. Our work instead concurrently learns a repertoire of diverse policies using QD algorithms and leverage the diversity of the behavioural repertoire as resets. In Multi-task MAP-Elites \\cite{mouret2020quality}, it is also showed that the behaviour space can also be viewed and formulated as a task-space where each cell is a task.\n\nIn the context of QD algorithms and behavioural repertoire learning, the Reset-Free Trial and Error (RTE) \\cite{chatzilygeroudis2018reset} algorithm has also aimed to address the reset problem. RTE demonstrated this for adaptation using the behavioural repertoire as a prior for Gaussian Process models. This is a different setting from the work we present in this paper as the behavioural repertoire generation process itself in RTE is performed fully in simulation using resets. The reset-free in RTE refers to the reset-free adaptation when performing sim-to-real transfer or reset-free adaptation to mechanical damage. In our work, we aim to learn the behavioural repertoire itself in a reset-free manner.\n\\section{Background: Dynamics-Aware QD} \\label{sec:background}\nWe build on the DA-QD framework proposed by Lim et al. \\cite{lim2021dynamics}. We briefly summarize DA-QD here and refer the reader to the full paper \\cite{lim2021dynamics} for further details. DA-QD is a model-based QD algorithm which extends the conventional QD framework \\cite{pugh2016quality, cully2017quality} discussed in section \\ref{subsec:rel-QD} with three key components: a \\textit{dynamics model}, an \\textit{imagined repertoire} and \\textit{selector} from the imagined repertoire.\n\nThe learnt dynamics model is a forward dynamics model and is represented by a neural network parameterized by $\\theta$. To capture both aleatoric and epistemic uncertainties, an ensemble of probabilistic models are used.\nThis forward model can be represented as $\\widetilde{p}_{\\vec \\theta}(\\vec s_{t+1} | \\vec s_t, \\vec a_t)$.\n\nHere, the disagreement between predictions of all models in the ensemble captures the epistemic uncertainty, i.e. it indicates the uncertainty of the prediction due to a lack of samples. The overall model disagreement $\\mu_d$ can be calculated as the expected difference between any two models in the ensemble $f_\\phi$ for one state-action pair, averaged over all time step predictions in one rolled-out trajectory (i.e. one evaluated behaviour) of length T~\\cite{Kidambi2020}:\n\\begin{equation} \\label{eqn:disagreement}\n \\begin{split}\n \\text{disag}(s,a) = \\mathop{\\mathbb{E}}_{i \\neq j} \\| f_{\\phi_i}(s,a) - f_{\\phi_j}(s,a) \\|_2 \\\\\n \\textstyle \\mu_d = \\frac{1}{T} \\sum_{t=0}^T \\text{disag}(s_t,a_t)\n \\end{split} \n\\end{equation}\nState transition data is collected and stored in a replay buffer $\\mathcal{B}$ as evaluations of robot behaviour are performed in the environment. The model is trained in a self-supervised manner to maximise log-likelihood of the transitions sampled from replay buffer and is optimized via back-propagation.\n\nThe dynamics model $\\widetilde{p}_{\\vec \\theta}$ can be called recursively to evaluate policies in what is referred to as an imagined roll-out. The expected fitness and BD can be obtained from this imagined roll-out as both these quantities measured are a function of the state trajectory. DA-QD introduced the concept of an imagined repertoire $\\smash{\\widetilde{\\mathcal{A}}}$ to organise and maintain solutions that have been evaluated in imagination using the dynamics model. The imagined repertoire $\\smash{\\widetilde{\\mathcal{A}}}$ uses the the same addition conditions as the repertoire $\\mathcal{A}$. The imagined repertoire $\\smash{\\widetilde{\\mathcal{A}}}$ only allows solutions that have been evaluated in imagination that are expected to be novel or better performing than existing solutions to be considered for evaluation. This is where the sample-efficiency of this method is derived from. Additionally, this allows QD to be performed fully in imagination. This means that the selection and mutation of the QD algorithm can be continuously performed from the imagined archive for any desired number of imagined generations without any samples or evaluations on the real system.\n\nFinally, with the introduction of the imagined repertoire $\\smash{\\widetilde{\\mathcal{A}}}$, this necessitates selection of solutions from $\\smash{\\widetilde{\\mathcal{A}}}$ to be evaluated. As the original DA-QD algorithm does not consider the reset-free sequential evaluation setting, the authors select all the solutions that have been added to the imagined archive to be evaluated in parallel. Our work extends DA-QD and proposes a more intelligent method to select and manage solutions in the imagined archive given the reset-free setting and safety constraints from the environment.\nAdditionally, DA-QD also does not explicitly use the resulting model-disagreement. In this paper, the model-disagreement is used both as a heuristic to select behaviours to execute more intelligently (Sec.~\\ref{subsec:methods-behaviour-selection}) and as an optimization objective in imagination (Sec. \\ref{sec:results_emitters}). \n\n\\section{Methods}\n\nWe present Reset-Free Quality-Diversity (RF-QD) as a method to enable the application of QD's behavioral repertoire learning in non-episodic real-world environments (see Algorithm \\ref{Algo:RF-QD_short}). We treat the robot as an actor in its environment that performs a constant search for new and improved behaviours and storing these in the archive. For this, we extend the classical QD loop by two steps. Firstly, we build on the pre-evaluation of any new behaviour \"in imagination\" by a dynamics model (DA-QD). Secondly, we introduce a behaviour selection policy, that modulates the robot's search for novel and high-performing behaviours as to comply with the safety constraints given by the environment (see Figure \\ref{fig:overview}).\n\nIn the following, we first elaborate on the core of our method: the \\textit{behaviour selection policy}. Then, we detail its main components: the \\textit{safety evaluation}, \\textit{safety constraints}, the \\textit{prioritisation metrics} and the \\textit{recovery policy}.\n\\input{pseudocode}\n\n\\subsection{Reset-free Behaviour Selection} \\label{subsec:methods-behaviour-selection}\nTo be able to stay safe while acting in its environment, we introduce a behaviour selection policy to modulate the robot's actions in the real world. This behaviour selection ensures that every new behaviour will only be performed if it is expected to be safe for the robot. \nAt every step, new behaviours are selected from a candidate buffer $\\mathcal{C}$. The candidate buffer\n$\\mathcal{C}$ is regularly filled with new policies from the imagined repertoire $\\smash{\\widetilde{\\mathcal{A}}}$ that are not already present in the repertoire $\\mathcal{A}$. \n$\\smash{\\widetilde{\\mathcal{A}}}$ is a component introduced in DA-QD that maintains solutions that were evaluated in imagination using the dynamics model $\\widetilde{p}_{\\vec \\theta}$.\nBased on the robot's current state in the environment, our policy then selects a subset of candidate behaviours $\\mathcal{C}_{safe}$ that have a low risk of violating the safety constraints given by the environment. Out of these, the candidate behaviour that has the highest projected prioritization score will then be evaluated in the real world. In the following sections, the core components will be described in more detail.\n\n\\subsection{Safety Evaluation}\nIn this paper, we assume knowledge of the environment layout, represented by 'safety regions' (see Figure \\ref{fig:illustration}), that indicate the region of dangerous states $\\Omega$. In practice, this information could as well be obtained using Simultaneous Localisation and Mapping (SLAM) methods with an on-board camera. \n\nDerived from the robot's state $s$, we define the exploration parameter $\\epsilon(s)$, which indicates the relative degree of safety in the current state. It is calculated as the smallest distance between $s$ and $\\Omega$ and normalised by the maximum encountered distance value (see Equation \\ref{eqn:epsilon_general}). While inside the safe region (i.e. $s \\notin \\Omega \\rightarrow \\epsilon(s) > 0$), the robot must choose any potential solution to be evaluated in the real world that is predicted to keep $\\epsilon(s) > 0$, i.e. does not enter any unsafe state.\nTo lower the risk of damage to the robot, an offset $\\beta$ can be added in the computation of $\\epsilon_s$ as an increased threshold for the minimum distance towards the border of the region of unsafe states within the state space.\n\n\\begin{equation}\n\\label{eqn:epsilon_general}\n \\epsilon(s) = \\frac{dist(s, \\Omega) - \\beta}\n {\\underset{s_i}{\\max} \\, dist(s_i, \\Omega) - \\beta}\n\\end{equation}\n\nFrom the dynamics model, we can obtain the predicted next state $s'$ after the execution of a candidate behaviour and compute $\\epsilon(s')$. $s'$ corresponds to the state $s_{t+T}$ after $T$ timesteps, where $T$ is the length of one behaviour.\nGenerally, we seek the robot to stay as close as possible to the safest point(s) in the environment, i.e. maintain maximal distance to the region of dangerous states ($\\epsilon(s) \\approx 1$).\n\n\\subsection{Safety Constraints}\nFor every behaviour selection performed by our policy, we first employ a safety constraint to determine the safe subset $\\mathcal{C}_{safe}$ of all available candidate behaviours with respect to the robot's current state. We can use different constraints depending on our knowledge of the environment and the intended risk aversion of our exploration. In the experiment section below, we evaluate the following constraints, all of which are based on the predicted robot state $s'$ after the execution of each imagined behaviour (given the current robot state $s$):\n\\begin{itemize}[leftmargin=*]\n\\setlength\\itemsep{1em}\n \\item As a \\textit{minimal constraint} we consider only candidate behaviours with $\\epsilon(s') > 0$ to ensure we never execute a behaviour that was already expected to be unsafe. \n\n \\item Alternatively, a \\textit{contextual constraint} carries weight only if the current robot state is near the border of the region of unsafe states ($\\epsilon(s) \\approx 0$), but enables free exploration if it is far away from potential danger ($\\epsilon(s) \\approx 1$):\n \\begin{equation}\n \\label{eqn:constraint_minimal}\n \\epsilon(s') > \\epsilon(s) \\cdot (1-\\epsilon(s))\n \\end{equation}\n \n \\item If we have access to the gradient of the epsilon function, the direction of maximal improvement of safety with respect to the next state can be computed as $\\nabla_s \\epsilon(s)$. The \\textit{gradient-minimal constraint} considers only solutions moving in the general direction of the gradient. Based on the dot product of the unit vectors of the gradient of the epsilon function ($\\nabla_s \\epsilon(s)$) and the projected movement in state space ($s'-s$), we formulate a lower bound for deviation from the direction of the gradient as:\n \\begin{equation}\n \\label{eqn:constraint_gradient_minimal}\n \\frac{s'-s}{||s'-s||}\\cdot\\frac{\\nabla_s \\epsilon(s)}{||\\nabla_s \\epsilon(s)||} \\geq 0\n \\end{equation}\n Geometrically, this is equal to a maximum deviation of 90\u00b0 in 2D space as visualised in Figure \\ref{fig:safety_constraint} (green semicircle).\n \n \\item Again, we can modify this into a more strict \\textit{gradient-contextual constraint} by using the value of epsilon at the current state of the robot to modulate the constraint. This way, the constraint is more relaxed towards the centre of the region of safe states but only accepts small deviations from the direction of the safety gradient close to the border of the region of unsafe states:\n \\begin{equation}\n \\label{eqn:constraint_gradient_contextual}\n \\frac{s'-s}{||s'-s||}\\cdot\\frac{\\nabla_s \\epsilon(s)}{||\\nabla_s \\epsilon(s)||}\n \\geq \\epsilon(s) \\cdot (1-\\epsilon(s))\n \\end{equation}\n Geometrically, this is equal to a deviation from the gradient proportional to $\\epsilon(s)$ (see yellow region in Figure \\ref{fig:safety_constraint} for $\\epsilon(s)=0.5$).\n \n \\item Finally, safety can also be enforced not by a hard constraint, but as a component of the prioritization measures. This can especially be useful as a supplement to the gradient-free constraints in complex environments.\n\\end{itemize}\n\n\\begin{figure}\n\\centering\n \\includegraphics[height=0.15\\textwidth]{figures\/constraint_grad_context_legend.png}\n \\caption{Sketch of the gradient-based safety constraints in a simple circular 2D-environment.}\n \\label{fig:safety_constraint}\n \\vspace{-4mm}\n\\end{figure}\n\n\\subsection{Prioritization Metrics}\nAfter the safe subset of candidate behaviours $\\mathcal{C}_{safe}$ has been selected based on the safety constraint, the remaining candidates are ranked according to a prioritization measure as the second step in behaviour selection. This is intended to give priority to the real-world evaluation of candidate behaviours which have the highest value for the overall QD algorithm performance, as real-world samples are expensive to collect. \nFinally, the candidate with the highest prioritization score is selected. The composition of prioritization measures can be adapted depending on the task at hand. We can either use a single prioritization measure or a (weighed) sum of multiple values. In this work, we have evaluated the following measures:\n\\begin{itemize}[leftmargin=*]\n\\setlength\\itemsep{1em}\n \\item Firstly, the robot's \\textbf{safety} can be considered again as a prioritization measure through the dynamic exploration parameter $\\epsilon(s')$ as outlined above. Generally, this approach will be used in combination with another metric to enable the behaviour policy to tolerate a possible safety violation in favor of a higher score.\n \n \\item Another key measure to score a candidate behaviour is the \\textbf{dynamics model disagreement}.\n The dynamics model used in DA-QD consists of an ensemble of models to capture the epistemic uncertainty via disagreement between the predictions of the models (see Section \\ref{sec:background} and Equation \\ref{eqn:disagreement}). \n The epistemic uncertainty can also be interpreted and formalised as an information theoretic measure of the expected information gain \\cite{pathak2019self, sekar2020planning}. \n Maximising the model-disagreement has been used as a self-supervised intrinsic reward for exploration in Deep RL literature \\cite{pathak2019self, sekar2020planning}. \n The key idea behind this measure is to prioritise policies that are most informative based on our current knowledge which is represented via the ensemble of dynamics models (i.e. epistemic uncertainty).\n Selecting policies with high-model disagreement would mean visiting states that have been less explored than others.\n As we incrementally train the dynamics model on incoming data, policies that visit states that have been seen will no longer have a large model disagreement which will allow this measure to continuously be used to explore.\n Depending on the state of the robot in the environment, we can prioritize high and low model disagreement behaviours. \n \n Conversely, policies with low disagreement should be prioritized in safety-critical situations. Solutions with low expected model disagreement are likely to resemble the expected outcome and indicates the model's confidence.\n \n \n \\item Finally, we also consider the classical metrics used to quantify behaviours in QD. This is firstly the \\textbf{novelty} of a candidate behaviour as the distance to the k nearest solutions already in the archive ($\\nu_1, ..., \\nu_k$)~\\cite{lehman2011abandoning}. Similarly, we could also consider the quality of a solution through a measure such as the QD improvement~\\cite{fontaine2020covariance} or the future value of a solution through its curiosity score~\\cite{cully2017quality}. However, this is left for future work.\n\\end{itemize}\n\n\\subsection{Recovery Policy}\nAs a final safeguard to keep the robot in the safe region of the environment, we introduce a recovery policy to return the robot to safety if it ever violates any of the environment's safety constraints. These constraints can be derived from the environment in various ways, e.g. as a minimum distance to obstacles represented by 'safety regions' as in this work. Should the robot leave the safe region, the discovery of new behaviours will be halted and a greedy behaviour selection policy will be employed over the archive of behaviours that were already evaluated in the environment instead of the buffer of candidate behaviours. Here, we pick the single behaviour that is projected to effect the greatest improvement in safety.\n\n\n\n\n\n\n\\section{Experiments}\nWe evaluate our method with an 18 DoF hexapod robot on an adapted version of the omni-directional locomotion task~\\cite{cully2013behavioral}.\nIn this task, the robot learns behaviours to walk in every direction from an initial position.\nFor the controllers, we evolve parameters of a sinusoidal control signal that is sent to each motor. This sinusoidal signal acts as a structural prior towards periodic movement for locomotion.\nAs we focus on a reset-free setting, all evaluations of new behaviours have to be done sequentially and cannot be parallelised.\nAll simulations are performed in RobotDART building on the Dynamics Animation and Robotics Tookit (DART) simulator \\cite{dartsim}. \nTo simulate a practical number of trials that would be performed in the real-world experiment, the number of evaluations performed in any single run of the algorithm are limited to 10,000.\n\n\\subsection{Baseline comparison}\n\\label{sec:results_baselines}\nFirstly, we evaluate the general capability of the RF-QD method.\nFor this, we compare against \"vanilla\" QD and DA-QD \\cite{lim2021dynamics} as baselines. RF-QD and both baselines use the Iso-dd~\\cite{vassiliades2018discovering} variation operator.\nWe use a simple flat environment with a circular region of safety with radius $r=2.0m$.\nFigure \\ref{fig:trajectory} shows example trajectories of the baselines compared to RF-QD. The baselines' random selection of behaviours causes the robot to trail off deeply into the dangerous region, while RF-QD performs its exploration almost entirely within the safe region. The depicted RF-QD run leaves the safe region once, but then deploys the recovery policy (blue line) to return to safety.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/figure4-rfqd.png}\n \\caption{Example trajectories of DA-QD, Vanilla-QD and RF-QD in flat environment with safe region (green) and dangerous region (red).}\n \\label{fig:trajectory}\n \\vspace{-4mm}\n\\end{figure}\n\nAs the baseline methods are not made for a reset-free environment, for all further comparisons we perform manual resets to the starting position if the robot leaves the safe region by more than 50 cm. This is similar to what is done when performing QD on a real-world robot today. For the baseline comparisons, RF-QD was run with a gradient-contextual safety constraint and encouraging maximal novelty through the prioritization strategy. This configuration has proven powerful in our evaluation of different constraints and prioritization measures. Table \\ref{tab:baselines_safety} quantifies the safety of the three algorithms averaged over 10 replications of each. We can see, that RF-QD achieves almost perfect safety - never once requiring a safety reset as described above and only rarely taking a single step outside the safe region.\n\n\\begin{table}\n \\caption{Safety metrics for all variants, averaged over 10 runs (mean \u00b1 std).}\n \\label{tab:baselines_safety}\n \\begin{tabular}{c|ccc}\n \\toprule\n Variant &Resets &Steps outside safety &Recovery steps\\\\\n \\midrule\n Vanilla-QD & 54.0 \u00b1 4.2 & 908.0 \u00b1 74.1 & n\/a \\\\\n DA-QD & 114.0 \u00b1 17.8 & 1039.5 \u00b1 51.0 & n\/a \\\\\n RF-QD & 0.0 \u00b1 0.0 & 1.0 \u00b1 2.8 & 3.5 \u00b1 9.9 \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nAdditionally, RF-QD slightly outperforms its direct baseline DA-QD in terms of both QD-score and coverage as shown in Figure \\ref{fig:baselines_performance}. While the distance to vanilla QD is due to DA-QD's increased sample efficiency, RF-QD's behaviour selection policy does not sacrifice performance for safety, but even improves performance by its candidate prioritization strategy (i.e. novelty in this case).\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/figure5-rfqd.png}\n \\caption{QD-Score and coverage of RF-QD and baselines on the circular safe area environment. The graphs represent the median as a coloured bold line, while the shaded area extends to the first and the third quartiles over 10 runs.}\n \\label{fig:baselines_performance}\n \\vspace{-4mm}\n\\end{figure}\n\n\n\\subsection{Comparison of Policy Configurations}\n\\label{sec:hyperparameters}\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.5\\textwidth]{figures\/figure6-rfqd.png}\n \\caption{Comparison of different Behaviour Selection Policy configurations on both performance (coverage) and safety (recovery steps) on the circular safe area environment.}\n \\label{fig:policy_comparison}\n \\vspace{-4mm}\n\\end{figure}\n\nAdditionally, we evaluated the various configurations of the Behaviour Selection Policy as introduced in Section \\ref{subsec:methods-behaviour-selection}. Figure \\ref{fig:policy_comparison} shows an overview over the different combinations of safety constraints and prioritization measures. Here, the policy configurations are evaluated by performance (represented by their final coverage) and safety (represented by the number of recovery steps), both from runs of 10,000 steps over 10 replications. In short, Figure \\ref{fig:policy_comparison} shows strong separation between the relatively unsafe minimal and contextual constraints (both gradient-free) and all remaining constraints. The strongest performance is exhibited by variants combining the novelty or disagreement maximising prioritization measures with a gradient contextual constraint. Out of the naive gradient-free constraints, which must be used if there is no single 'safest' direction of movement (as e.g. in more complex environments such as the one following in Section \\ref{sec:results_complex}), only the soft constraints achieves comparable safety scores and performances as the gradient-based configurations. Which exact configuration should be chosen will however always depend on the exact task at hand.\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=1\\textwidth]{figures\/figure7-rfqd.png}\n \\caption{Complex environments with 0, 5, 10 and 15 obstacles. Top: Example trajectories of hexapod acting under RF-QD. Middle: Example archives by RF-QD. Bottom: Example archives by DA-QD.}\n \\label{fig:environments}\n \\vspace{-2mm}\n\\end{figure*}\n\n\\subsection{Robustness to environment complexity}\n\\label{sec:results_complex}\nTo evaluate RF-QD's performance in increasingly complex environments, we exchange the previous circular environment for a closed 4x4m room with a number of column-shaped obstacles. Figure \\ref{fig:environments} shows examples of such environments including RF-QD's trajectories in them (top row). \nWe can observe that the robot acting under RF-QD keeps its distance from the obstacles, while building archives of behaviours (middle row) that are radically less affected by the environment complexity than those created by DA-QD (bottom row).\n\nIn these complex environments, we employed RF-QD with a safety-focused configuration. This uses a minimal (hard) safety constraint combined with two equally weighed prioritization measures to select behaviours that maximise safety (through $\\epsilon$) and have low model disagreement. \nAs a benchmark for QD performance, we again add a version of DA-QD that uses safety resets, now triggered on any collision with an obstacle. \nWe also keep a 'naive' version of DA-QD, that is not reset upon collision (same as RF-QD). These algorithms were compared in rooms with 0 to 15 obstacles (see Figure \\ref{fig:advanced_qd_scores}). \nWhile in an empty room, all algorithms perform similarly well, the naive DA-QD variant quickly drops in performance with a growing number of obstacles through a large number of collisions (which render the corresponding evaluations invalid). \nAt the same time, RF-QD manages to fully keep up with the upper baseline of DA-QD (using safety resets). \nWhile a more performance-focused prioritization strategy (i.e. novelty as in Section \\ref{sec:results_baselines}) for RF-QD might have increased QD-scores slightly, this would have sacrificed the safety of the robot in more challenging environments.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.5\\textwidth]{figures\/figure8-rfqd.png}\n \\caption{Increasingly complex environments: QD-Scores vs number of obstacles. The graphs represent the mean as a coloured bold line, while the shaded area extends to the standard deviations over 10 runs for each environment.}\n \\label{fig:advanced_qd_scores}\n \\vspace{-4mm}\n\\end{figure}\n\n\n\\subsection{Effect of objectives in imagination} \\label{sec:results_emitters}\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.5\\textwidth]{figures\/rfqd-figure9.pdf}\n \\caption{Study of different optimization objectives and prioritization metric configurations. Each panel considers a different prioritisation metric. Top: Disagreement of selected behaviours by RF-QD. The bold lines and shaded areas represent the median and interquartile range over 10 replications respectively. Middle: Progression of the archive size over the number of selected behaviours for each optimization objective. Bottom: Distribution of the total number of recovery steps for each optimization objective.}\n \\label{fig:emitter_results}\n \\vspace{-4mm}\n\\end{figure}\nWe also study the effect of the type of solutions available in the candidate buffer that the behaviour selection policy chooses from.\nTo study this, we investigate the influence of different optimisation objectives for the generation of the candidate buffer during the QD in imagination. \nWhen using Iso-DD \\cite{vassiliades2018discovering}, the solutions are relatively generic and objective-agnostic, i.e., not optimised to fulfil a specific objective. \nAlternatively, we can use different types of emitters (introduced by CMA-ME \\cite{fontaine2020covariance}) to produce solutions that maximise a specific objective.\nWe perform experiments using three different optimization objectives: maximising model disagreement, minimising model disagreement, and a random direction objective as a surrogate objective for novelty. We compare this to the standard Iso-dd variations used in all our experiments as a baseline.\nWe perform an ablation of these three different objectives with their corresponding prioritization measures used in the behaviour selection policy. We report results across 10 replications.\n\nFirst, we evaluate the effect of more targeted objectives by analysing the model disagreement associated with the individuals selected by the behaviour selection policy (Figure \\ref{fig:emitter_results}). \nThe key take-away from Figure \\ref{fig:emitter_results} (top) is that the optimisation objectives used when running QD in imagination can strongly influence the behaviours that are finally selected. \nWe can see that regardless of the prioritization metric used by the behaviour selection policy, the same overall trends are always observed:\nThe minimising disagreement optimization objective (yellow) always results in low disagreement individuals being selected by the behaviour selection policy regardless of the prioritization metrics.\nThe same observation applied to the maximising disagreement objective (green).\nThis observation corresponds to our initial hypothesis where targeted optimization objectives can skew the distribution of solutions generated towards the target objective.\nThis results in a higher probability for the solutions with the desired metric being selected.\n\nGiven that biased\/specialised sets of solutions can be generated in the candidate buffer using more targeted objectives, we evaluate the effect of the composition of this candidate buffer on the performance of RF-QD. \nFigure \\ref{fig:emitter_results} (middle and bottom) show that the objective-agnostic Iso-DD operator outperforms all the targeted optimization objectives both in terms of coverage and safety (number of resets) across all prioritization measures used by the behaviour selection policy.\nThis is an interesting result as one could expect the variants with aligned prioritization measures and optimization objectives to perform better.\nWe hypothesize that the buffer of candidate solutions being generated by targeted objectives become too specialised while the objective-agnostic Iso-DD can generate a diverse buffer of solutions to choose from.\nThis is not such a surprising observation as Multi-Emitter MAP-Elites \\cite{cully2020multi} had previously also shown that when using simultaneously multiple emitter types, the random emitter (based on Iso-dd) remains the most fruitful through the entire process compared to other objective-driven emitters.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion} \\label{label:discussion}\nIn this paper, we have presented RF-QD, a method to learn behavioural repertoires autonomously without resets in realistic environments. We demonstrate how an intelligent behaviour selection policy can be used with QD in imagination to learn safely and efficiently. We first test RF-QD to learn while remaining within a designated area and show that the behaviour selection policy is necessary to prevent the need for resets and to stay within the safe training area. \nWe then show how RF-QD can also operate in more complex environments with many obstacles and minimal room for error.\nOur results also show that we can acquire full repertoires despite increasing environment complexity while the performance of DA-QD and Vanilla QD baselines deteriorate with the increase in complexity. \nLastly, we conduct an ablation to investigate the effect of the type of solutions present in the candidate buffer on the performance of RF-QD.\nWe demonstrate that using targeted optimization objectives when performing QD in imagination can bias the distribution of solutions presented to the behaviour selection policy. \nOur results show that it is important to keep the diverse types of solutions in the candidate buffer over just specialised solutions biased towards a single metric.\n\nFor future work, we also hope to show RF-QD learning directly on a real world system, with no dependence on simulators. Additionally, this paper only considers safety and danger in the form of obstacle avoidance. We leave other forms dangerous scenarios and work on safety detection for future work.\n\n\n\n\n\n\n\\section{Research Methods}\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:Intro}\n\nGabor systems are fundamental objects in time-frequency analysis. Given a set $\\Lambda \\subset \\mathbb{R}^{2l}$ and a function $g \\in L^2(\\mathbb{R}^l)$, the Gabor system $G(g, \\Lambda)$ is defined as \n\\begin{eqnarray*} G(g,\\Lambda) &=& \\{ g(x-m) e^{2\\pi i n\\cdot x}\\}_{(m,n) \\in \\Lambda}. \\end{eqnarray*}\nWhen $\\Lambda$ is taken to be $\\mathbb{Z}^{2l}$, $G(g)=G(g,\\mathbb{Z}^{2l})$ is referred to as the \\emph{integer lattice Gabor system} generated by $g$. The Balian-Low theorem (BLT) and its generalizations are uncertainty principles concerning the generator $g$ of such a system in the case that $G(g,\\Lambda)$ forms a Riesz basis. \n\\begin{theorem}[BLTs]\\label{thm:BLT}\n\tLet $g\\in L^2(\\mathbb{R})$ and suppose that the Gabor system $G(g)=G(g,\\mathbb{Z}^2)$ is a Riesz basis for $L^2(\\mathbb{R})$. \n\t\\begin{enumerate}\n\t\t\\item[(i)] If $10$, depending only on $A$ and $B$, such that for any $N\\ge 2$ and for any $b\\in \\ell_2^d$ which generates an $A,B$-Gabor Riesz basis for $\\ell_2^d$, \n\t\\begin{eqnarray*} c_{AB}\\log(N) &\\le& \\|N\\Delta b\\|_{\\ell_2^d}^2 + \\|N \\Delta \\mathcal{F}_d(b)\\|_{\\ell_2^d}^2. \\end{eqnarray*}\n\tConversely, there exists a constant $C_{AB}$ such that for any $N\\ge2$, there exists $b\\in \\ell_2^d$ which generates and $A,B$-Gabor Riesz basis for $\\ell_2^d$ such that \n\t\\begin{eqnarray*} \\|N\\Delta b\\|_{\\ell_2^d}^2 + \\|N \\Delta \\mathcal{F}_d(b)\\|_{\\ell_2^d}^2 &\\le& C_{AB}\\log(N). \\end{eqnarray*}\n\\end{theorem}\n\n\nNitzan and Olsen also show that the continuous BLT, Theorem \\ref{thm:BLT}, follows from this discrete version and that the following \\emph{Finite Quantitative BLT} also holds.\n\\begin{theorem}[Theorem 5.3, \\cite{Nitzan}]\\label{thm:FiniteQuantBLT}\n\tLet $A,B>0$. There exists a constant $C_{AB}>0$ such that the following holds. Let $N \\ge 200\\sqrt{B\/A}$ and let $b \\in \\ell_2^d$ generate an $A,B$-Gabor Riesz basis. Then, for all positive integers $1\\le Q, R\\le (N\/16)\\sqrt{A\/B}$, we have \n\t\\begin{eqnarray*} \\frac{1}{N} \\sum_{j=NQ}^{d-1} |b(j)|^2 + \\frac{1}{N} \\sum_{k=NR}^{d-1} |\\mathcal{F}_d b(k)|^2 &\\ge& \\frac{C_{AB}}{QR}. \\end{eqnarray*}\n\\end{theorem}\n\n\n\\subsection{Extension to Several Variables}\nThe first goal of this paper is to extend Theorem \\ref{thm:FiniteBLT} and \\ref{thm:FiniteQuantBLT} to several variables, which we state below in Theorems \\ref{thm:FiniteBLTHD} and \\ref{thm:FiniteQuantBLTHD}. \n\nWe consider complex-valued sequences on $\\mathbb{Z}_d^l=\\mathbb{Z}_d \\times \\cdots \\times \\mathbb{Z}_d$ for $l\\ge 1$, and we denote the set of all such sequences as $\\ell_2^{d,l}$. The view of these sequences as samples of a continuous $g \\in L^2([-\\frac{N}{2},\\frac{N}{2}]^l)$, where $b(\\v{j})=g(\\v{j}\/N)$ for $\\v{j}=(j_1,...,j_l)\\in I_d^l$ leads to the normalization \n\\begin{eqnarray*} \\|b\\|_{\\ell_2^{d,l}}^2 \\ =\\ \\frac{1}{N^l} \\sum_{\\v{j} \\in \\mathbb{Z}_d^l} |b(\\v{j})|^2 \\ =\\ \\frac{1}{N^l} \\sum_{\\v{j} \\in I_d^l} |b(\\v{j})|^2. \\end{eqnarray*}\nThe discrete Fourier transform, $\\mathcal{F}_{d,l}$, on $\\ell_2^{d,l}$, is given by \n\\begin{eqnarray*} \\mathcal{F}_{d,l}(b)(\\v{k}) &=& \\frac{1}{N^l} \\sum_{\\v{j}\\in \\mathbb{Z}_d^l} b(\\v{j}) e^{-2\\pi i \\frac{\\v{j}\\cdot \\v{k}}{d}}. \\end{eqnarray*}\nUnder this normalization, $\\mathcal{F}_{d,l}$ is an isometry on $\\ell_2^{d,l}$. The Gabor system generated by $b$, $G_{d,l}(b)$ is given by \n\n\\begin{eqnarray*} G_{d,l}(b) \\ =\\ \\{ b(\\v{j}-N\\v{n}) e^{2\\pi i \\frac{\\v{j}\\cdot \\v{m}}{N}}\\}_{(\\v{n},\\v{m})\\in \\{0,...,N-1\\}^{2l}}\\ =\\ \\{ b(\\v{j}-\\v{n})e^{2\\pi i \\frac{\\v{j}\\cdot \\v{m}}{d}}\\}_{(\\v{n},\\v{m})\\in (N\\mathbb{Z}_d)^{2l}}. \\end{eqnarray*} \n\nFor any $k \\in \\{1,...,l\\}$, let $\\Delta_k: \\ell_2^{d,l}\\rightarrow \\ell_2^{d,l}$ be defined by \n\\begin{eqnarray*} \\Delta_k b (\\v{j}) &=& b(\\v{j}+\\v{e}_k)-b(\\v{j}), \\end{eqnarray*} \nwhere $\\{\\v{e}_k\\}_{k \\in \\{1,...,l\\}}$ is the standard orthonormal basis for $\\mathbb{R}^l$. Then $N \\Delta_k b$ approximates the partial derivative $\\frac{\\partial g}{\\partial x_k}$. \n\n\n\nWe have the following generalization of Theorem \\ref{thm:FiniteBLT}. \n\\begin{theorem} \\label{thm:FiniteBLTHD}\n\tFix constants $00$, we let $\\{ |j_k|\\ge t\\}$ denote the set $\\{ \\v{j} \\in I_d^l: |j_k|\\ge t\\}$. \n\n\n\n\\begin{theorem}\\label{thm:FiniteQuantBLTHD}\n\tLet $A, B>0$ and $l \\in \\mathbb{N}$. There exists a constant $C>0$ depending only on $A$, $B$, and $l$, such that the following holds. Let $N\\ge 200\\sqrt{B\/A}$ and let $b \\in \\ell_2^{d,l}$ generate an $A,B$-Gabor Riesz basis for $\\ell_2^{d,l}$. Then, for any $1 \\le k \\le l$ and all integers $1\\le Q, R\\le (N\/16) \\sqrt{A\/B}$, we have \n\t\\begin{eqnarray*} \\frac{1}{N^l} \\sum_{|j_k|\\ge \\frac{NR}{2}} |b(\\v{j})|^2 + \\frac{1}{N^l} \\sum_{|j_k|\\ge \\frac{NQ}{2}} |\\mathcal{F}_{d,l}b (\\v{j})|^2 &\\ge& \\frac{C}{QR}. \\end{eqnarray*}\n\\end{theorem}\n\n\n\n\n\\subsection{Finite Nonsymmetric BLTs}\nIn Section \\ref{FQBLTHDapps}, we prove nonsymmetric versions of the finite BLT. In the process, we show that symmetric and nonsymmetric versions of the finite BLT follow as corollaries of the finite quantitative BLT (Theorem \\ref{thm:FiniteQuantBLTHD}), as long as $N$ is sufficiently large.\n\n\n\n\\begin{theorem}[Nonsymmetric Finite BLT]\\label{thm:NonsymFiniteBLT}\n\tLet $A,B>0$ and $10$, depending only on $A, B, p$ and $q$ such that the following holds. Let $N \\ge 200\\sqrt{B\/A}$. Then, for any $b\\in \\ell_2^{d,l}$ which generates an $A,B$-Gabor Riesz basis for $\\ell_2^{d,l}$, \n\t\\begin{eqnarray*} C\\log(N) &\\le& \\frac{1}{N^{l}} \\sum_{\\v{j} \\in I_{d}^l} \\left|\\frac{j_k}{N}\\right|^p |b(\\v{j})|^2+ \\frac{1}{N^{l}} \\sum_{\\v{j} \\in I_{d}^l} \\left|\\frac{j_k}{N}\\right|^q |b(\\v{j})|^2. \\end{eqnarray*} \n\\end{theorem}\n\n\\begin{remark}\n\tTheorem \\ref{thm:NonsymFiniteBLT} gives a finite dimensional version of the nonsymmetric BLT for parameters satisfying $11$, then \n\t\t\\begin{eqnarray*} \\frac{C}{\\tau-1} &\\le& \\int_{\\mathbb{R}^l} |x_k|^p |g(x)|^2 dx + \\int_{\\mathbb{R}^l} |\\xi_k|^q |\\widehat{g}(\\xi)|^2d\\xi.\\end{eqnarray*}\n\t\\end{enumerate}\n\\end{theorem}\n\nWhen the bound $2\\le T <\\infty$ is replaced by $1<\\gamma \\le T<\\infty$, the bound $\\frac{C(1-2^{\\tau-1})}{(1-\\tau)} T^{1-\\tau}$ in part \\textit{(i)} can be replaced by $\\frac{C(1-\\gamma^{\\tau-1})}{(1-\\tau)} T^{1-\\tau}$. In Section \\ref{FQBLTHDapps} we extend this theorem to the case where either $p=\\infty$ or $q=\\infty$.\n\nThe first and second inequalities in Theorem \\ref{thm:QuantBLTCorollaryNonsymmetric} quantify the growth of `localization' quantities in terms of cutoff weights of the form $\\min(|x_k|^p, T)$. The $\\log$ term in the second inequality shows a connection between the continuous BLT and its finite dimensional versions. The last inequality, on the other hand, shows that generators of Gabor Riesz bases must satisfy a Heisenberg type uncertainty principle for every $0< p \\le 2$. A similar inequality is known to hold for arbitrary $L^2(\\mathbb{R})$ functions by a result of Cowling and Price \\cite{CP}. However, for generators of Gabor Riesz bases, we have explicit estimates on the dependence of the constant on $\\tau$ and the result here is stated for higher dimensions.\n\n\n\n\n\n\n\n\n\\section{Preliminaries: The Zak Transform and Quasiperiodic Functions} \\label{xcom2}\n\nThe Zak transform is an essential tool for studying lattice Gabor systems. The discrete Zak transform $Z_{d,l}$ of $b\\in \\ell_2^{d,l}$ for $(\\v{m},\\v{n}) \\in \\mathbb{Z}_d^{2l}$ is given by\n\n\\begin{align*} Z_{d,l}(b)(\\v{m}, \\v{n}) &= \\sum_{\\v{j} \\in \\{0,...,N-1\\}^l} b(\\v{m}-N\\v{j}) e^{2\\pi i \\frac{\\v{n}\\cdot \\v{j}}{N}}=\\sum_{\\v{j} \\in N\\mathbb{Z}_d^l} b(\\v{m}-\\v{j})e^{2\\pi i \\frac{\\v{n} \\cdot \\v{j}}{d}}. \\end{align*} \n\n\n\nThe following properties show that $Z_{d,l}(b)$ encodes basis properties of $G_{d,l}(b)$, while retaining information about `smoothness' (see the remark following Proposition \\ref{prop:ZakProperties}) of $b$ and $\\mathcal{F}_{d,l}(b)$. Note that $Z_{d,l}(b)(\\v{m},\\v{n})$ is defined for $(\\v{m},\\v{n})\\in \\mathbb{Z}_d^{2l}$ and is $d$-periodic in each of its $2l$ variables. However, the Zak transform satisfies even stronger periodicity conditions. \nIn fact, $Z_{d,l}(b)$ is \\emph{$N$-quasiperiodic on $\\mathbb{Z}_d^{2l}$}, that is\n\\begin{eqnarray}\nZ_{d,l}(b) (\\v{m}+N\\v{e}_k, \\v{n})&=& e^{2\\pi i \\frac{n_k}{N}} Z_{d,l}(b)(\\v{m},\\v{n}), \\label{eqn:quasiperiodic}\\\\\nZ_{d,l}(b) (\\v{m}, \\v{n}+N\\v{e}_k)&=& Z_{d,l}(b)(\\v{m},\\v{n}). \\nonumber\n\\end{eqnarray}\nLet $S_N=\\{0,...,N-1\\}$. Then, the quasi-periodicity conditions above show that $Z_{d,l}(b)$ is completely determined by its values on $S_N^{2l}$. \n\nWe will use the notation $\\ell_2(S_N^{2l})$ to denote the set of sequences $W(\\v{m},\\v{n})$ defined on $S_N^{2l}$ with norm given by \n\\[ \\|W\\|_{\\ell_2(S_N^{2l})}^2 \\ =\\ \\frac{1}{N^{2l}} \\sum_{(\\v{m}, \\v{n}) \\in S_N^{2l}} |W(\\v{m},\\v{n})|^2,\\]\nwhere here we keep the variables $\\v{m}$ and $\\v{n}$ separate due to the connection with the Zak transform. The normalization is chosen so that if $W$ is a sampling of a function $h(\\v{x}, \\v{y})$ on $[0,1]^{2l}$, then $\\|W\\|_{\\ell_2(S_N^{2l})}$ approximates the $L^2([0,1]^{2l})$ norm of $h$. \n\nThe Zak transform has many other important properties, some of which we collect in the next proposition. Arguments for these facts are standard and presented in \\cite{AGT} and \\cite{Nitzan}, for instance.\n\\begin{proposition}\\label{prop:ZakProperties}\n\tLet $b \\in \\ell_2^{d,l}$. \n\t\\begin{enumerate}\n\t\t\\item[(i)] $Z_{d,l}$ is a unitary mapping from $\\ell_2^{d,l}$ onto $\\ell_2(S_N^{2l})$. \n\t\t\\item[(ii)] A sequence $b\\in \\ell_2^{d,l}$ generates an $A,B$-Gabor Riesz basis for $\\ell_2^{d,l}$ if and only if $Z_{d,l}(b)$ satisfies \n\t\t\\begin{eqnarray*} A \\ \\le\\ |Z_{d,l}(b)(\\v{m}, \\v{n}) |^2 \\ \\le\\ B, \\text{ for } (\\v{m}, \\v{n}) \\in \\mathbb{Z}_d^{2l}. \\end{eqnarray*}\n\t\t\\item[(iii)] Let $\\widehat{b}\\ =\\ \\mathcal{F}_{d,l}(b)$. Then,\n\t\t\\begin{eqnarray*} Z_{d,l}(\\widehat{b})(\\v{m},\\v{n})\\ =\\ e^{2\\pi i \\frac{\\v{m}\\cdot \\v{n}}{d}} Z_{d,l}(b)(-\\v{n},\\v{m}). \\end{eqnarray*}\n\t\t\\item[(iv)] For $a,b \\in \\ell_2^{d,l}$ define $(a \\ast b)(\\v{k})= \\frac{1}{N^l} \\sum_{j \\in \\mathbb{Z}_d^{l}} a(\\v{k}-\\v{j})b(\\v{j})$. Then, \n\t\t\\begin{eqnarray*} Z_{d,l}(a\\ast b)(\\v{m}, \\v{n})\\ =\\ \\frac{1}{N^l} \\sum_{\\v{j} \\in \\mathbb{Z}_d^l} b(\\v{j}) Z_{d,l}(a)(\\v{m}-\\v{j},\\v{n}) \\ =\\ (Z_{d,l}(a)\\ast_1 b) (\\v{m},\\v{n}), \\end{eqnarray*}\n\t\twhere $\\ast_1$ denotes convolution of $b$ with respect to the first set of variables of $Z_{d,l}(a)$, $\\v{m}$, keeping the second set, $\\v{n}$, fixed.\n\t\t\n\t\\end{enumerate}\n\\end{proposition}\n\n\\begin{remark} We will be interested in the `smoothness' of $b$ and $Z_{d,l}(b)$ for $b \\in \\ell_2^{d,l}$. Since these are functions on discrete sets, smoothness is not well defined, but we use the term in relation to the size of norms of certain difference operators defined on $\\ell_2^{d,l}$ and $\\ell_2(S_N^{2l})$, which mimic norms of partial derivatives of differentiable functions. \\end{remark}\n\nFor $1\\le k \\le l$ and any $N$-quasiperiodic function on $\\mathbb{Z}_d^l$, let $\\Delta_k,\\Gamma_k$ be defined as follows:\n\\begin{eqnarray*}\n\t\\Delta_k W(\\v{m},\\v{n}) &=& W(\\v{m}+\\v{e_k},\\v{n})-W(\\v{m},\\v{n}),\\\\\n\t\\Gamma_k W(\\v{m},\\v{n}) &=& W(\\v{m},\\v{n}+\\v{e_k})-W(\\v{m},\\v{n}).\n\\end{eqnarray*}\nFor $b \\in \\ell_2^{d,l}$ define $\\alpha_k(b)$ and $\\beta_k(b)$ by\n\n\\begin{eqnarray*}\n\t\\alpha_k(b)&=& \\|N\\Delta_k b\\|_{\\ell_2^{d,l}}^2 + \\|N \\Delta_k \\mathcal{F}_{d,l}(b)\\|_{\\ell_2^{d,l}}^2,\\\\\n\t\\beta_k(b)&=& \\frac{1}{N^{2l}} \\sum_{(\\v{m},\\v{n})\\in S_N^{2l}} |N\\Delta_k Z_{d,l}(b)(\\v{m},\\v{n})|^2+\\frac{1}{N^{2l}} \\sum_{(\\v{m},\\v{n})\\in S_N^{2l}} |N\\Gamma_k Z_{d,l}(b)(\\v{m},\\v{n})|^2.\n\\end{eqnarray*}\n\nThe following proposition shows that $\\alpha_k(b)$ and $\\beta_k(b)$ are essentially equivalently sized. Proposition 4.1 in \\cite{Nitzan} proves this for the case $l=k=1$, and it is readily checked that the proof carries over directly to the $l>1$ setting. \n\\begin{proposition}\\label{prop:alphabeta}\n\tLet $B>0$ and let $b\\in \\ell_2^{d,l}$ be such that $|Z_{d,l}(b)(\\v{m},\\v{n})|^2 \\le B$ for all $(\\v{m},\\v{n})\\in \\mathbb{Z}_d^{2l}$. Then, for all integers $N\\ge 2$ and any $1 \\le k \\le l$, we have \n\t\\begin{eqnarray*} \\frac{1}{2} \\beta_{k}(b) -8\\pi^2 B\\ \\le\\ \\alpha_{k}(b) \\ \\le\\ 2 \\beta_{k}(b) + 8 \\pi^2 B. \\end{eqnarray*}\n\\end{proposition}\nWe thus see that in order to bound $\\alpha_k(b)$ as in Theorem \\ref{thm:FiniteBLTHD}, it is sufficient to bound $\\beta_k(b)$. For $b \\in \\ell_2^d= \\ell_2^{d,1}$, let $\\beta(b)= \\beta_{1}(b)$, and let \n\\begin{eqnarray*}\n\t\\beta_{A,B}(N)=\\inf\\{ \\beta(b)\\},\n\\end{eqnarray*}\nwhere the infimum is taken over all $b \\in \\ell_2^d$ such that $b$ generates an $A,B$-Gabor Riesz basis. \n\\begin{theorem}[Theorem 4.2, \\cite{Nitzan}]\\label{thm:betabound1d}\n\tThere exist constants $00$ such that for any $N\\ge 2$, there exists a $b \\in \\ell_2^d$ such that $G_d(b)$ is an orthonormal basis for $\\ell_2^d$ and \n\t\\[ \\beta(b)\\ =\\ \\sum_{(m,n)\\in S_N^2} \\left|\\Delta Z_{d,1}(b)(m,n) \\right|^2+\\sum_{(m,n)\\in S_N^2} \\left|\\Gamma Z_{d,1}(b)(m,n) \\right|^2 \\le C \\log (N).\\]\n\t\n\tFor $\\v{j} \\in \\mathbb{Z}_d^l$, let $b_l(\\v{j})= b(j_1) b(j_2)\\cdots b(j_l)$. Then,\n\t\\[ Z_{d,l}(b_l)(\\v{m},\\v{n})\\ =\\ Z_{d,1}(b)(m_1, n_1) \\cdots Z_{d,1}(b)(m_l,n_l).\\]\n\tSince $G_d(b)$ is an orthonormal basis for $\\ell_2^d$, $Z_{d,l}(b_{l})$ is unimodular, and therefore, $G_{d,l}(b_l)$ is an orthonormal basis for $\\ell_2^{d,l}$ by Proposition \\ref{prop:ZakProperties}. We have, $\\beta_{1}(b_l)$ is equal to\n\t\n\t\\begin{gather*} \\frac{1}{N^{2(l-1)}} \\sum_{(\\v{m}',\\v{n}')\\in \\mathbb{Z}_N^{2(l-1)}} \\left[\\sum_{(m_1,n_1)\\in S_N^2} \\left|\\Delta Z_{d,1}(b)(m_1,n_1) \\right| ^2 + \\sum_{(m_1,n_1)\\in S_N^2} \\left|\\Gamma Z_{d,1}(b)(m_1,n_1) \\right| ^2\\right] \\\\\n \\le C\\log(N). \\end{gather*}\n\\end{proof}\n\n\nTheorem \\ref{thm:FiniteBLTHD} follows by combining Theorem \\ref{thm:betaboundhd} with Proposition \\ref{prop:alphabeta}.\n\n\n\n\n\\section{Proof of Theorem \\ref{thm:FiniteQuantBLTHD}} \\label{sec:proofFiniteQuantBLTHD}\n\n\nIn establishing a Finite Quantitative BLT for several variables, we follow a similar argument used to prove the one variable version (from \\cite{Nitzan}), but there are some necessary updates to certain parts of the proof. We include the details here for completeness. \n\nWe start with a straightforward bound on the `smoothness' of $Z_{d,l}(b\\ast \\phi)$. This observation is analogous to Lemma 2.6 of \\cite{Nitzan}. Let $\\|\\phi\\|_{\\ell_1^{d,l}} = \\frac{1}{N^l} \\sum_{\\v{j}\\in \\mathbb{Z}_d^l} |\\phi(\\v{j})|$, and for $a, b \\in \\ell_2^{d,l}$, recall that $(a \\ast b)(\\v{k})= \\frac{1}{N^l} \\sum_{\\v{j} \\in \\mathbb{Z}_d^l} a(\\v{k}-\\v{j})b(\\v{j})$. \n\\begin{lemma}\\label{lem:convboundzd}\n\tSuppose $b, \\phi \\in \\ell_2^{d,l}$ are such that $|Z_{d,l}(b)|^2 \\le B$ everywhere. Then, for any integer $t$, \n\t\\begin{eqnarray*}\n\t\t|Z_{d,l}(b\\ast \\phi) (\\v{m}+t\\v{e}_k,\\v{n})-Z_{d,l}(b\\ast \\phi) (\\v{m}, \\v{n})| \\le \\frac{\\sqrt{B} |t|}{N} \\|N \\Delta_k \\phi\\|_{\\ell_1^{d,l}}.\n\t\\end{eqnarray*}\n\\end{lemma}\n\n\\begin{proof}\n\tFrom Proposition \\ref{prop:ZakProperties}, we have \n\t\\begin{eqnarray*}\n\t\tZ_{d,l}(b\\ast \\phi)(\\v{m}, \\v{n}) = \\frac{1}{N^l} \\sum_{\\v{j} \\in \\mathbb{Z}_d^l} \\phi(\\v{j}) Z_{d,l}(b)(\\v{m}-\\v{j}, \\v{n})= Z_{d,l}(b) \\ast_1 \\phi(\\v{m},\\v{n}).\n\t\\end{eqnarray*}\n\tTherefore, we have \n\t\\begin{eqnarray*}\n\t\t& & |Z_{d,l}(b\\ast \\phi) (\\v{m}+t\\v{e}_k,\\v{n})-Z_{d,l}(b\\ast \\phi) (\\v{m}, \\v{n})|\\\\\n\t\t&\\le& \\sum_{s=0}^{t-1} |Z_{d,l}(b\\ast \\phi) (\\v{m}+(s+1)\\v{e}_k,\\v{n})-Z_{d,l}(b\\ast \\phi) (\\v{m}+s\\v{e}_k, \\v{n})|\\\\\n\t\t&=& \\sum_{s=0}^{t-1} \\left| \\frac{1}{N^l} \\sum_{\\v{j}\\in \\mathbb{Z}_d^l} Z_{d,l}(b)(\\v{j}, \\v{n}) [ \\phi (\\v{m}+(s+1)\\v{e}_k -\\v{j}) -\\phi (\\v{m}+s\\v{e}_k -\\v{j})] \\right|\\\\\n\t\t&\\le& \\sum_{s=0}^{t-1} \\frac{\\sqrt{B}}{N^l} \\sum_{\\v{j}\\in \\mathbb{Z}_d^l} |\\Delta_k \\phi (\\v{m}+s\\v{e}_k - \\v{j})|\\ =\\ \\frac{\\sqrt{B}}{N} t \\|N\\Delta_k \\phi\\|_{\\ell_1^{d,l}}.\n\t\\end{eqnarray*} \n\\end{proof}\n\nNext we extend the following Lemma 5.2 of \\cite{Nitzan} to higher dimensions. The adjustments to this lemma for the higher dimensional setting are minimal, however we state the one-dimensional and multi-variable versions separately for comparison.\n\n\\begin{lemma}[Lemma 5.2, \\cite{Nitzan}]\\label{conv-lemma}\nLet $A, B>0$ and $N\\geq 200 \\sqrt{B\/A}$. There exist positive constants $\\delta=\\delta(A)$ and $C=C(A,B)$ such that the following holds (with $d=N^2$). Let\n\\begin{itemize}\n\\item[(i)] \\quad $Q, R \\in \\mathbb{Z}$ such that $1\\leq Q,R \\leq (N\/16) \\cdot \\sqrt{A\/B}$,\n\\item[(ii)] \\quad $\\phi,\\psi \\in \\ell_2^d$ such that $\\sum_n|\\Delta\\phi(n)|\\leq 10 R$ and $\\sum_n|\\Delta\\psi(n)|\\leq 10 Q$,\n\\item[(iii)] \\quad $b\\in \\ell_2^d$ such that $A \\leq |Z_d (b)|^2 \\leq B$.\n\\end{itemize}\nThen, there exists a set $S\\subset ([0,N-1]\\cap\\mathbb{Z})^2$ of size $|S|\\geq C N^2\/ QR$\n such that all $(u,v)\\in S$ satisfy either\n\\begin{align}\\label{conv-ineq-1}\n|Z_d(b)(u,v)-Z_d(b\\ast \\phi)(u,v)|\\geq\\delta, \\qquad \\text{or} \\\\[2mm]\n\\label{conv-ineq-2}\n|Z_d( \\mathcal{F}_d b)(u,v)-Z_d(( \\mathcal{F}_d b)\\ast \\psi)(u,v)|\\geq\\delta.\n\\end{align}\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lem:generatorConvSmooth}\n\tLet $A,B>0$, $1\\le k \\le l$, and $N\\ge 200\\sqrt{B\/A}$. There exist positive constants $\\delta=\\delta(A)$ and $C=C(A, B)$, such that the following holds. Let \n\t\\begin{enumerate}\n\t\t\\item[(i)] $Q, R \\in \\mathbb{Z}$ be such that $1\\le Q, R \\le \\frac{N}{16} \\sqrt{\\frac{A}{B}}$\n\t\t\\item[(ii)] $\\phi, \\psi \\in \\ell_2^{d,l}$ be such that $\\|N\\Delta_k \\phi\\|_{\\ell_1^{d,l}} \\le 10 R$ and $\\|N\\Delta_k \\psi\\|_{\\ell_1^{d,l}} \\le 10 Q$\n\t\t\\item[(iii)] $b \\in \\ell_2^{d,l}$ be such that $A\\le |Z_{d,l}(b)|^2 \\le B$. \n\t\\end{enumerate}\n\tThen, there exists a set $S\\subset ([0,N-1]\\cap \\mathbb{Z})^{2l}$ of size $|S|\\ge CN^{2l}\/Q R$ such that all $(\\v{u}, \\v{v}) \\in S$ satisfy either \n\t\\begin{eqnarray}\n\t|Z_{d,l}(b) (\\v{u}, \\v{v}) - Z_{d,l}(b\\ast \\phi) (\\v{u}, \\v{v}) |&\\ge& \\delta, \\text{ or}\\label{eqn:first} \\\\\n\t|Z_{d,l}(\\mathcal{F}_{d,l}b) (\\v{u}, \\v{v}) - Z_{d,l}((\\mathcal{F}_{d,l}b)\\ast \\psi) (\\v{u}, \\v{v}) |&\\ge& \\delta. \\label{eqn:second}\n\t\\end{eqnarray}\n\\end{lemma}\n\n\n\n\\begin{proof}\t\t\n\tWithout loss of generality, we prove this for $k=1$. \n\t\n\tAs in Lemma 5.2 of \\cite{Nitzan}, let $\\delta_1= 2\\sqrt{A} \\sin(\\pi ( \\frac{1}{4}-\\frac{1}{200}))$. Also, choose $K$ and $L$ to be the smallest integers satisfying \n\t\\[ \\frac{200 \\sqrt{B} R}{9\\delta_1}\\le K \\le N\\ \\ \\ \\ \\text{and}\\ \\ \\ \\ \\frac{\\sqrt{B}}{\\delta_1} \\max\\left\\{ \\frac{200 Q}{9}, 80 \\pi\\right\\} \\le L \\le N.\\]\n\t\n\tFor $s, t \\in \\mathbb{Z}$, let \n\t\\[ \\sigma_s=\\left[ \\frac{sN}{K}\\right],\\ \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\ \\omega_t=\\left[ \\frac{tN}{L}\\right], \\]\n\tand let $\\Sigma=\\inf_s \\{ \\sigma_{s+1}-\\sigma_s\\}\\ge \\left[\\frac{N}{K}\\right] \\ge \\frac{N}{K}$, $\\Omega = \\inf_t\\{ \\omega_{t+1}-\\omega_t\\}\\ge \\frac{N}{L}$. Then, we have \n\t\\[ \\Sigma \\Omega \\ge C_1 \\frac{N^2}{QR},\\]\n\twhere $C_1$ can be chosen to be \n\t\\[ C_1= \\left[(\\frac{200 \\sqrt{B}}{9\\delta_1}+1)(\\frac{\\sqrt{B}}{\\delta_1} \\max(\\frac{200}{9},80\\pi) +1)\\right]^{-1}.\\]\n\t\n\tWe recall the following definition from \\cite{Nitzan}. For $(u,v) \\in ([0,\\Sigma-1] \\cap \\mathbb{Z}) \\times ([0,\\Omega-1]\\cap \\mathbb{Z})$, let \n\t\\[ \\text{Lat}(u,v)=\\{(u+\\sigma_s,v+\\omega_t):(s,t)\\in ([0,K-1]\\cap \\mathbb{Z})\\times ([0,L-1]\\cap \\mathbb{Z})\\},\\]\n\tand \n\t\\[ \\text{Lat}^*(u,v)=\\{(N-v-\\omega_t,u+\\sigma_s):(s,t)\\in ([0,K-1]\\cap \\mathbb{Z})\\times ([0,L-1]\\cap \\mathbb{Z})\\}.\\]\n\tNote that $\\text{Lat}(u,v)$ and $\\text{Lat}(u',v')$ are disjoint for distinct $(u,v)$ and $(u',v')$, and similarly for $\\text{Lat}^*(u,v)$. However, it is possible that $\\text{Lat}(u,v) \\cap \\text{Lat}^*(u',v') \\neq \\emptyset$ for some $(u,v)$ and $(u',v')$.\n\t\n\tNow similarly, for any $(\\v{m}', \\v{n}') \\in ([0,N-1]\\cap \\mathbb{Z})^{2(l-1)}$, let\n\t\\[ \n\t\\text{Lat}_{(\\v{m}',\\v{n}')}(u,v)= \\{ ((m_1,\\v{m}'),(n_1,\\v{n}')): (m_1,n_1)\\in \\text{Lat}(u,v)\\},\\] \n\tand\n\t\\begin{eqnarray*}\n\t\t\\text{Lat}^*_{(\\v{m}',\\v{n}')}(u,v)= \\{ ((n_1,N-\\v{n}'),(m_1,\\v{m}')):(n_1,m_1)\\in \\text{Lat}^*(u,v) \\}.\n\t\\end{eqnarray*}\n\tHere, by $N-\\v{n'}$ we mean $(N-n'_1, N-n'_2,..., N-n'_{l-1})$. We have that $\\text{Lat}_{(\\v{m}',\\v{n}')}(u,v) \\cap \\text{Lat}_{(\\v{m}'',\\v{n}'')}(u',v')=\\emptyset$ unless it holds that $((u,\\v{m}'),(v,\\v{n}'))=$ $ ((u',\\v{m}''),(v',\\v{n}''))$, and similar properties for $\\text{Lat}^*_{(\\v{m}',\\v{n}')}(u,v)$.\n\t\n\tNow, fix $(\\v{m}', \\v{n}') \\in ([0,N-1]\\cap \\mathbb{Z})^{2(l-1)}$, and consider \\[T(m_1, n_1)=T_{\\v{m}',\\v{n}'}(m_1,n_1)=Z_{d,l}(b)( (m_1,\\v{m}'), (n_1, \\v{n}')),\\] for $(m_1, n_1) \\in \\mathbb{Z}_d^2$. Note that $T$ is $N$-quasiperiodic on $\\mathbb{Z}_d^2$, and satisfies $A\\le |T|^2 \\le B$. \n\t\n\tFor each $(u,v)\\in([0,\\Sigma-1] \\cap \\mathbb{Z}) \\times ([0,\\Omega-1]\\cap \\mathbb{Z})$, Corollary 3.6 of \\cite{Nitzan} guarantees at least one point $(s,t) \\in ([0,K-1] \\cap \\mathbb{Z}) \\times ([0,L-1]\\cap \\mathbb{Z})$ so that either \n\t\\begin{eqnarray}\n\t&|T(u+\\sigma_{s+1}, v+\\omega_t) - T(u+\\sigma_s,v+\\omega_t)| \\ge \\delta_1, \\text{ or} \\label{eqn:third}\\\\\n\t&|T(u+\\sigma_s, v+\\omega_{t+1}) - T(u+\\sigma_s,v+\\omega_t)| \\ge \\delta_1. \\label{eqn:fourth}\n\t\\end{eqnarray}\n\nWe now make a claim which will furnish the last part of the proof of the lemma.\n\t\n\t\\begin{claim}\n\t\tFor $u$, $v$, $\\sigma_s$, $\\omega_t$, $\\v{m}'$ and $\\v{n}'$ as above, \n\t\t\\begin{enumerate}\n\t\t\t\\item[(i)] If \\eqref{eqn:third} is satisfied, then there exists $(\\v{a},\\v{b})\\in \\text{Lat}_{(\\v{m}',\\v{n}')}(u,v)$ so that \\eqref{eqn:first} is satisfied for $\\delta=\\frac{\\delta_1}{20}$. \n\t\t\t\\item[(ii)] If \\eqref{eqn:fourth} is satisfied, then there exists $(\\v{a},\\v{b})\\in \\text{Lat}^*_{(\\v{m}',\\v{n}')}(u,v)$ so that \\eqref{eqn:second} is satisfied for $\\delta= \\frac{\\delta_1}{40}$\n\t\t\\end{enumerate}\n\t\\end{claim}\n\t\n\tBefore proving this claim, we show how to complete the proof of the lemma. For a fixed $(\\v{m}', \\v{n}')$ there are $\\Sigma \\Omega \\ge C_1 \\frac{N^2}{QR}$ distinct choices of $(u,v)$ to consider and each of them either falls in part (\\textit{i}) or (\\textit{ii}) of the claim. Let $S^1_{(\\v{m}',\\v{n}')}$ be the set of $(u,v)$ points which fall into category (\\textit{i}), and similarly let $S^2_{(\\v{m}', \\v{n}')}$ be the set of $(u,v)$ points which fall into category (\\textit{ii}). Then, for either $i=1,2$, we must have \n\t\\begin{equation}\\label{eqn:Sbound} \n\t|S^i_{(\\v{m}', \\v{n}')} |\\ \\ge\\ \\frac{C_1N^2}{2QR}.\n\t\\end{equation}\n\t\n\tNow, there are $N^{2(l-2)}$ possible choices of $(\\v{m}', \\v{n}')$. Let $S_1$ be the set of all $(\\v{m}', \\v{n}')$ such that \\eqref{eqn:Sbound} is satisfies with $i=1$, and let $S_2$ be the set of all $(\\v{m}', \\v{n}')$ such that \\eqref{eqn:Sbound} is satisfied with $i=2$. So at least one of $S_1$ or $S_2$ must contain $N^{2(l-2)}\/2$ elements. \n\t\n\tIn the case that $S_1$ contains this many elements (the $S_2$ case is nearly identical and left to the reader), since $\\text{Lat}_{(\\v{m}', \\v{n}')}(u,v)$ are disjoint for distinct $((u,\\v{m}'),(v,\\v{n}'))$, we find at least $\\frac{C_1N^{2l}}{4QR}= C\\frac{N^{2l}}{QR}$ distinct points all satisfy \\eqref{eqn:first} if $i=1$. The lemma is then proved conditioning on the claim above. We then establish finally the two part claim. \\\\\n\n\t\n\t\\textit{Proof of Claim.}\n\tFor both parts we use properties of the Zak transform detailed in Proposition \\ref{prop:ZakProperties}. First we show part \\textit{i)}. Let $H(u,v)=Z_{d,l}(b\\ast \\phi)((u,\\v{m}'),(v,\\v{n}'))$. Note that Lemma \\ref{lem:convboundzd} and the assumptions on $R$ and $\\|N \\Delta_1 \\phi\\|_{\\ell_1^{d,l}}$ imply that for any integer $t$ satisfying $t\\le \\frac{2N}{K}$, \n\t\\begin{eqnarray}\n\t|H(u+t,v)-H(u,v)|&\\le \\frac{2\\sqrt{B} }{K} \\|N \\Delta_1 \\phi\\|_{\\ell_1^{d,l}}\\le \\frac{ 20\\sqrt{B} R}{K}\\le \\frac{9\\delta_1}{10}. \\label{eqn:Hbound}\n\t\\end{eqnarray}\n\tSo, if \\eqref{eqn:third} is satisfied, using \\eqref{eqn:Hbound}, we have\n\t\\begin{eqnarray*}\n\t\t\\delta_1 &\\le& |T(u+\\sigma_{s+1}, v+\\omega_t) - T(u+\\sigma_s,v+\\omega_t)|\\\\\n\t\t&\\le& |T(u+\\sigma_{s+1}, v+\\omega_t) - H(u+\\sigma_{s+1}, v+\\omega_t)| \\\\\n\t\t& &\\ \\ \\ \\ \\ + \\frac{9\\delta_1}{10}+|T(u+\\sigma_{s}, v+\\omega_t) - H(u+\\sigma_{s}, v+\\omega_t)|.\n\t\\end{eqnarray*}\n\tUpon rearranging terms, we find \n\t\\begin{eqnarray*}\n\t\t\\frac{\\delta_1}{10} &\\le |T(u+\\sigma_{s+1}, v+\\omega_t) - H(u+\\sigma_{s+1}, v+\\omega_t)|+|T(u+\\sigma_{s}, v+\\omega_t) - H(u+\\sigma_{s}, v+\\omega_t)|,\n\t\\end{eqnarray*}\n\twhich shows that \\eqref{eqn:first} is satisfied for $\\delta'= \\frac{\\delta_1}{20}$, and for either $((u+\\sigma_{s+1},\\v{m}'),(v+\\omega_t,\\v{n}'))$ or $((u+\\sigma_{s},\\v{m}'),(v+\\omega_t,\\v{n}'))$. If $(u+\\sigma_{s+1}, v+\\omega_t)$ is not in $\\text{Lat}(u,v)$, by the N-quasiperiodicity of $T$, we may find another point in $\\text{Lat}(u,v)$ which satisfies the same bound. \n\t\n\tNow we prove part \\textit{ii)}. Letting $\\hat{b}= \\mathcal{F}_{d,l}(b)$, we have,\n\t\\begin{eqnarray*}\n\t\t\\delta_1&\\le& |T(u+\\sigma_s, v+\\omega_{t+1}) - T(u+\\sigma_s,v+\\omega_t)| \\\\\n\t\t&=& |Z_{d,l}(b)((u+\\sigma_s, \\v{m}'),(v+\\omega_{t+1},\\v{n}')) - Z_{d,l}(b)((u+\\sigma_s, \\v{m}'),(v+\\omega_{t},\\v{n}'))|\\\\\n\t\t&=& |Z_{d,l}(\\hat{b})((-v-\\omega_{t+1},-\\v{n}'),(u+\\sigma_s, \\v{m}'))\\\\\n\t\t&\\ &\\ - e^{-2\\pi i (\\omega_{t+1}-\\omega_t)(u+\\sigma_s) \/d} Z_{d,l}(\\hat{b})((-v-\\omega_{t},-\\v{n}'),(u+\\sigma_s, \\v{m}'))|\\\\\n\t\t&=& |Z_{d,l}(\\hat{b})((N-v-\\omega_{t+1},N-\\v{n}'),(u+\\sigma_s, \\v{m}'))\\\\\n\t\t&\\ &\\ - e^{-2\\pi i (\\omega_{t+1}-\\omega_t)(u+\\sigma_s) \/d} Z_{d,l}(\\hat{b})((N-v-\\omega_{t},N-\\v{n}'),(u+\\sigma_s, \\v{m}'))|,\n\t\\end{eqnarray*}\n\twhere we have used that $Z_{d,l}(b)(\\v{m},\\v{n})= e^{2\\pi i \\v{m}\\cdot \\v{n}\/d} Z_{d,l}(\\widehat{b})(-\\v{n},\\v{m})$ in the second step, and for the last step we have used $N$-quasiperiodicity. \n\t\n\tLet $\\widetilde{T}(v,u)=Z_{d,l}(\\hat{b})((v,N-\\v{n}'),(u, \\v{m}'))$, \n\tand $\\widetilde{H}(v,u)= Z_{d,l}(\\hat{b}\\ast \\psi)((v,N-\\v{n}'),(u, \\v{m}'))$.\n\tThen, \n\t\\begin{eqnarray*}\n\t\t\\delta_1 & \\le & |\\widetilde{T}(N-v-\\omega_{t+1},u+\\sigma_s)- e^{-2\\pi i (\\omega_{t+1}-\\omega_t)(u+\\sigma_s) \/d}\\widetilde{T}(N-v-\\omega_{t},u+\\sigma_s)|\\\\\n\t\t& \\le & | \\widetilde{T}(N-v-\\omega_{t+1},u+\\sigma_s)- \\widetilde{T}(N-v-\\omega_{t},u+\\sigma_s)| +\\frac{\\delta_1}{20}. \n\t\\end{eqnarray*}\n\tCombining these, we see that \n\t\\begin{eqnarray*}\n\t\t\\frac{19}{20} \\delta_1 &\\le& | \\widetilde{T}(N-v-\\omega_{t+1},u+\\sigma_s)- \\widetilde{T}(N-v-\\omega_{t},u+\\sigma_s)|. \n\t\\end{eqnarray*}\n\tArguing as in the first case above, and replacing $H$ by $\\widetilde{H}$ and $T$ by $\\widetilde{T}$, we find that either $((N-v-\\omega_{t+1},N-\\v{n}'),(u, \\v{m}'))$, or $((N-v-\\omega_{t},N-\\v{n}'),(u, \\v{m}'))$ satisfy \\eqref{eqn:second}, with $\\delta= \\frac{\\delta_1}{40}$. Again, using quasi-periodicity, we can guarantee that there is a point in $\\text{Lat}^*_{(\\v{m}',\\v{n}')}(u,v)$ satisfying \\eqref{eqn:second}. \n\t\n\\end{proof} \\vspace{2 mm}\n\n\n\nFinally, we follow the construction of \\cite{Nitzan} to create the functions $\\phi$ and $\\psi$ appearing in the previous lemma (Lemma \\ref{lem:generatorConvSmooth}) which in turn are used to prove Theorem \\ref{thm:FiniteQuantBLTHD}. Let $\\rho$ : $\\mathbb{R}\\rightarrow \\mathbb{R}$ be the inverse Fourier transform of \n\\[\\hat{\\rho}(\\xi)= \\begin{dcases} \\hspace{15 mm} 1,\\hspace{10 mm} |\\xi|\\leq 1\/2 \\\\ 2(1-\\xi sgn(\\xi)),\\hspace{5 mm} 1\/2\\leq |\\xi|\\leq 1 \\\\ \\hspace{15 mm} 0,\\hspace{10 mm} |\\xi|\\geq1\\end{dcases}. \\] \n\nFor $f \\in L^2(\\mathbb{R})$ satisfying $\\sup_{t \\in \\mathbb{R}} |t^2 f(t)|<\\infty$ and $\\sup_{\\xi \\in \\mathbb{R}} |\\xi^2 \\widehat{f}(t)|<\\infty$, let \n\\[ P_N f(t)\\ =\\ \\sum_{k=-\\infty}^\\infty f(t+kN)\\]\nand for an $N$-periodic continuous function $h$, let \n\\[ S_N h\\ =\\ \\{ h(j\/N)\\}_{j =0}^{d-1}.\\]\n\n\nLet $\\rho_R(t)=R\\rho(Rt)$. Fix $1\\le k \\le l$, and for $\\v{j} \\in I_d^l$ define the vector $\\v{j}'=(j_1,..., j_{k-1},j_{k+1},...,j_l) \\in I_{d}^{l-1}$, and let \n\\[ \\phi_{R,k}(\\v{j}) \\ =\\ N^{l-1} \\delta_{\\v{j}', \\v{0}} \\left(S_N P_N \\rho_{R} (j_k) \\right).\\]\nNow $\\phi_{R,k} (\\v{j})$ is equal to $ \\left(S_N P_N \\rho_{R} (j_k) \\right)$ when $j_i=0$ for each $i \\neq k$, and is zero otherwise. \n\\begin{lemma}\n\tLet $\\phi_{R,k}$ be as above for a positive integer $R$. Then,\n\t\\[ \\|N \\Delta_k \\phi_{R,k}\\|_{\\ell_1^{d,l}} \\ \\le\\ 10 R.\\]\n\\end{lemma}\n\\begin{proof}\n\tWe have \n\t\\begin{eqnarray*}\n\t\t\\|N \\Delta_k \\phi_{R,k}\\|_{\\ell_1^{d,l}}&=& \\frac{1}{N^l} \\sum_{\\v{j} \\in I_d^l} N| \\Delta_k \\phi_{R,k} (\\v{j})| \\ =\\ \\sum_{j_k \\in I_d} |\\Delta S_N P_N \\rho_{R}(j_k)|.\n\t\\end{eqnarray*}\n\tLemma 2.10 and Lemma 5.1 of \\cite{Nitzan} show that the right hand side is bounded by $10R$. \n\\end{proof}\n\nWe now have sufficient tools to prove the Finite Quantitative BLT, Theorem \\ref{thm:FiniteQuantBLTHD}.\n\n\\begin{proof}[Theorem \\ref{thm:FiniteQuantBLTHD}]\n\tFor simplicity we show the result for $k=1$. \n\tLet $R$ and $Q$ be integers such that $1 \\le R, Q \\le (N\/16)\\sqrt{A\/B}$. Let $\\phi= \\phi_{R,1}$ and $\\psi=\\phi_{Q,1}$, and note that Lemma \\ref{lem:convboundzd} shows that \n\t\\[ \\|N\\Delta_1\\phi \\|_{\\ell_1^{d,l}} \\le 10 R, \\text{ and } \\|N\\Delta_1\\psi \\|_{\\ell_1^{d,l}} \\le 10 Q.\\]\n\tProposition 2.8 of \\cite{Nitzan}, and the fact that $\\mathcal{F}_d(N\\delta_{j,0})(k)=1$ for all $k \\in I_d$, shows that \n\t\\begin{align}\n\t\\mathcal{F}_{d,l} (\\phi)(\\vec{k}) &= \\mathcal{F}_d(S_N P_N \\rho_{R})(k_1) \\nonumber \\\\\n\t\t& = (S_N P_N \\mathcal{F}(\\rho_{R})) (k_1) = (S_N P_N \\widehat{\\rho}(\\cdot\/R)) (k_1),\\label{eqn:phifunction}\n\t\\end{align}\n\tand since $R1$}. Finally, in this case\n\n\t\\begin{eqnarray*}\n\t\t\\frac{C}{\\tau-1}&=&C\\int_1^\\infty S^{-\\tau} dS\\\\\n\t\t&\\le& \\int_{\\mathbb{R}^{l-1}} \\int_0^\\infty \\int\\limits_{|x_1|\\ge S^{1\/p}} |g(x_1,x')|^2 dx_1 dS dx'+ \\int_{\\mathbb{R}^{l-1}} \\int_0^\\infty \\int\\limits_{|\\xi_1|\\ge S^{1\/q}} |\\widehat{g}(\\xi_1,\\xi')|^2 d\\xi_1 dS d\\xi'\\\\\n\t\t&=& \\int_{\\mathbb{R}^{l}} |x_1|^p |g(x)|^2 dx+ \\int_{\\mathbb{R}^{l}} |\\xi_1|^q |\\widehat{g}(\\xi)|^2 d\\xi.\n\t\\end{eqnarray*}\n\n\\end{proof}\n\nThe following result generalizes part (ii) of Theorem \\ref{thm:BLT}. \n\\begin{theorem}\\label{thm:compactFunctionQuantCorr}\n\tSuppose $1\\le p < \\infty$, and $g \\in L^2(\\mathbb{R}^l)$ is such that $G(g)= \\{e^{2\\pi i n \\cdot x} g(x-m)\\}_{(m,n) \\in \\mathbb{Z}^{2l}}$ is a Riesz basis for $L^2(\\mathbb{R}^l)$ and $g$ is supported in $(-M,M)^l$. Then, there exists a constant $C$ depending only on the Riesz basis bounds of $G(g)$ such that for any $1\\le k \\le 1$ and any $2 \\le T \\le \\infty$ each of the below hold.\n\t\\begin{enumerate}\n\t\t\\item[(i)] If $p>1$, then \n\t\t\\[ \\frac{C(1-2^{1\/p-1})}{M(1-1\/p)}\\ \\le\\ \\int_{\\mathbb{R}^l} \\min(|\\xi_k|^p, T) |\\widehat{g}(\\xi)|^2 d\\xi.\\]\n\t\t\\item[(ii)] If $p = 1$, then \n\t\t\\[ \\frac{C\\log(T)}{M}\\ \\le\\ \\int_{\\mathbb{R}^l} \\min(|\\xi_k|, T) |\\widehat{g}(\\xi)|^2 d\\xi.\\]\n\t\t\\item[(iii)] If $p<1$, then \n\t\t\\[ \\frac{C}{M(1\/p-1)}\\ \\le\\ \\int_{\\mathbb{R}^l} |\\xi_k|^p, |\\widehat{g}(\\xi)|^2 d\\xi.\\]\n\t\\end{enumerate}\n\tThis result also holds when $g$ and $\\widehat{g}$ are interchanged.\n\\end{theorem}\nThe proof is nearly identical to that of Theorem \\ref{thm:QuantBLTCorollaryNonsymmetric}, after noticing that by applying the quantitative BLT with $R=M$, the integral related to $|g(x)|^2$ is zero due to the support assumption. Note that letting $T \\rightarrow \\infty$ in part (ii) gives part (ii) of Theorem \\ref{thm:BLTHD}.\n\n\nFinally, we focus on the finite nonsymmetric BLT. For $1\\le p,q<\\infty$ and $b \\in \\ell_2^{d,l}$, let \n\\[ \\alpha_k^{p,q}(b)\\ =\\ \\frac{1}{N^{l}} \\sum_{\\v{j} \\in \\mathbb{Z}_{d}^l} \\left|\\frac{j_k}{N}\\right|^p |b(\\v{j})|^2+ \\frac{1}{N^{l}} \\sum_{\\v{j} \\in \\mathbb{Z}_{d}^l} \\left|\\frac{j_k}{N}\\right|^q |\\mathcal{F}_{d,l}b(\\v{j})|^2.\\] To give a finite dimensional analog of part (ii) of Theorem \\ref{thm:BLT}, it will be convenient to define $\\alpha^{p,\\infty}_{k}(b)$ and $\\alpha^{\\infty, q}_k(b)$ as\n\\[\\alpha^{p,\\infty}_{k}(b)\\ =\\ \\frac{1}{N^{l}} \\sum_{\\v{j} \\in \\mathbb{Z}_{d}^l} \\left|\\frac{j_k}{N}\\right|^p |b(\\v{j})|^2, \\ \\ \\alpha^{\\infty,q}_k(b)\\ =\\ \\frac{1}{N^{l}} \\sum_{\\v{j} \\in \\mathbb{Z}_{d}^{l}} \\left|\\frac{j_k}{N}\\right|^q |\\mathcal{F}_{d,l}b(\\v{j})|^2.\\]\n\n\n\\begin{theorem}\\label{thm:NonsymFiniteBLTwithInfinity}\n\tLet $A,B>0$ and $1\\le p,q< \\infty$ and let $\\tau=\\frac{1}{p}+\\frac{1}{q}$. Assume $b\\in \\ell_2^{d,l}$ generates an $A, B$-Gabor Riesz basis for $\\ell_2^{d,l}$. There exists a constant $C>0$, depending only on $A, B, p$ and $q$ such that the following holds. Let $N \\ge 200\\sqrt{B\/A}$. \n\t\\begin{enumerate}\n\t\t\\item[(i)] If $\\tau=\\frac{1}{p}+\\frac{1}{q}<1$,\n\t\t\\begin{eqnarray*} C \\frac{N^{1-\\tau}}{1-\\tau} &\\le& \\alpha^{p,q}_{k}(b).\\end{eqnarray*}\n\t\t\\item[(ii)] If $\\tau=\\frac{1}{p}+\\frac{1}{q}=1$, \n\t\t\\begin{eqnarray*} C \\log(N) &\\le& \\alpha^{p,q}_{k}(b.)\\end{eqnarray*}\n\t\t\\item[(iii)] If $\\tau=\\frac{1}{p}+\\frac{1}{q}>1$,\n\t\t\\begin{eqnarray*} C\\frac{1-(200\/16)^{1-\\tau}}{\\tau-1} &\\le& \\alpha^{p,q}_{k}(b).\\end{eqnarray*}\n\t\\end{enumerate}\n\tAlso, if $\\mathcal{F}_{d,l}(b)$ is supported in the set $(-\\gamma_N N\/2,\\gamma_N N\/2)\\cap \\mathbb{Z}$ where $\\gamma_N= \\lfloor (N\/16)\\sqrt{A\/B}\\rfloor$, then parts (i), (ii), and (iii) hold with $\\tau=\\frac{1}{p}$ and $\\alpha^{p,q}(b)$ replaced by $\\alpha^{p,\\infty}(b)$. Similarly, if $b$ is supported in the set $(-\\gamma_N N\/2,\\gamma_N N\/2)\\cap \\mathbb{Z}$ then parts (i), (ii), and (iii) hold with $\\tau=\\frac{1}{q}$ and $\\alpha^{p,q}(b)$ replaced by $\\alpha^{\\infty, q}(b)$. \n\\end{theorem}\n\n\nWe start with a lemma giving a bound on a typical sum arising in the proof which follows. Similar to above, $\\{ b>|j_k| \\ge a\\}$ will be used to denote $\\{ \\v{j} \\in I_d^l: b>|j_k| \\ge a\\}$. \n\\begin{lemma}\\label{lem:BasicSumBounds}\n\tLet $1\\le \\nu<\\infty$, $N>200 \\nu$, $c=1\/(16\\nu)$, and $\\gamma_N=\\lfloor c N\\rfloor$. If $0<\\alpha\\le 1$, then for any $b \\in \\ell_2^{d,l}$, we have\n\t\\[ \\sum_{S=1}^{\\gamma_N} \\sum_{|j_k|\\ge NS^\\alpha\/2} |b(\\v{j})|^2 \\ \\le\\ 2^{1\/\\alpha} \\sum_{\\v{j} \\in \\mathbb{Z}^d} \\left|\\frac{j_k}{N}\\right|^{1\/\\alpha} |b(\\v{j})|^2,\\]\n\twhere $C_\\alpha$ only depends on $\\alpha$. \n\\end{lemma}\nNote, we will apply this lemma with $\\nu= \\sqrt{B\/A}$ where $A$ and $B$ are Riesz basis bounds of $G_{d,l}(b)$ for some $b \\in \\ell_2^{d,l}$. However, this lemma holds regardless of whether $G_{d,l}(b)$ is basis for $\\ell_2^{d,l}$. \n\\begin{proof}\n\tRearranging terms, we have\n\n\t\\begin{eqnarray}\n\t\\sum_{S=1}^{\\gamma_N} \\sum_{|j_k|\\ge NS^\\alpha\/2} |b(\\v{j})|^2 = \\sum_{m=1}^{\\gamma_N-1} m \\sum_{\\frac{N(m+1)^\\alpha}{2}> |j_k|\\ge \\frac{Nm^\\alpha}{2}} |b(\\v{j})|^2 + \\gamma_N \\sum_{|j_k|\\ge \\frac{N \\cdot \\gamma_N^\\alpha}{2}} |b(\\v{j})|^2. \\label{eqn:CantComeUpWithGoodName}\n\t\\end{eqnarray}\n\n\tNote that for some $m$, if $j_k$ satisfies $|j_k|\\ge \\frac{Nm^\\alpha}{2}$, then $m \\le 2^{1\/\\alpha} \\left| \\frac{j_k}{N}\\right|^{1\/\\alpha}$.\n\tThen, from \\eqref{eqn:CantComeUpWithGoodName}, we find\n\t\\begin{eqnarray*}\n\t\t\\sum_{S=1}^{\\gamma_N} \\sum_{|j_k|\\ge NS^\\alpha\/2} |b(\\v{j})|^2 &\\le& 2^{1\/\\alpha}\\sum_{m=1}^{\\gamma_N-1} \\sum_{\\frac{N(m+1)^\\alpha}{2}> |j_k|\\ge \\frac{Nm^\\alpha}{2}} \\left| \\frac{j_k}{N}\\right|^{1\/\\alpha}|b(\\v{j})|^2 \\\\ &\\ +& 2^{1\/\\alpha} \\sum_{|j_k|\\ge \\frac{N \\gamma_N^\\alpha}{2}} \\left| \\frac{j_k}{N}\\right|^{1\/\\alpha}|b(\\v{j})|^2\\\\\n\t\t&\\le& 2^{1\/\\alpha}\\sum_{\\v{j} \\in I_d^l} \\left |\\frac{j_k}{N}\\right|^{1\/\\alpha} |b(\\v{j})|^2.\n\t\\end{eqnarray*} \n\\end{proof}\n\n\n\n\n\n\\begin{proof}[Theorem \\ref{thm:NonsymFiniteBLTwithInfinity}]\n\tWe prove the result for $k=1$. We treat the case where $p$ and $q$ are both finite and the case where one of these is infinite separately. Below, we take $\\tau=\\frac{1}{p}+\\frac{1}{q}$. \n\t\n\t\\textbf{Case 1: $1\\le p, q <\\infty$}. Let $S$ be an integer satisfying $1\\le S \\le \\gamma_N$ where $\\gamma_N = \\lfloor (N\/16)\\sqrt{A\/B}\\rfloor$, and $R= \\lceil S^{1\/p}\\rceil$, $Q=\\lceil S^{1\/q}\\rceil$ if $11\n\t\\end{cases},\n\t\\end{equation}\n\twhere the constants $C_{\\tau,A,B}$ depend only on $\\tau$, $A$, and $B$. \n\t\n\t\n\t\n\t\\textbf{Case 2: One of $p$ or $q$ is $\\infty$}. We can assume without loss of generality that $q=\\infty$ and $1\\le p<\\infty$. With this in mind, assume $b$ generates an $A,B$-Gabor Riesz basis for $\\ell_2^{d,l}$, and further suppose $\\mathcal{F}_{d,l}(b)$ is supported in the set $(-\\gamma_N N\/2, \\gamma_N N\/2)\\cap \\mathbb{Z}$. Then, Theorem \\ref{thm:FiniteQuantBLTHD} applied with $Q=\\gamma_N$, gives \n\t\\[ \\frac{C}{R\\gamma_N} \\ \\le\\ \\frac{1}{N^l} \\sum_{|j_k|\\ge \\frac{NR}{2}} |b(\\v{j})|^2,\\]\n\twhere the second sum does not appear due to the support condition on $\\mathcal{F}_{d,l}(b)$. As in part (i), let $1 \\le S \\le \\gamma_N$ and $R=\\lceil S^{1\/\\alpha}\\rceil$ if $1 0$, $b_1 > 0$, and $W_t\\sim i.i.d.~N(0,1)$. In other words, a potentially qualified transformation related to the GARCH(1,1) or ARCH($\\infty$) model can be exhibited as:\n\\begin{equation}\n W_t = \\frac{Y_t}{\\sqrt{a + a_1Y_{t-1}^2 + b_1\\sigma_{t-1}^2}} \\label{eq:3.2}\n\\end{equation}\nHowever, recall the core insight of the NoVaS method is connecting the original data with the transformed data by a qualified transformation function. A primary problem is desired to be solved is that the right-hand side of \\cref{eq:3.2} contains other terms rather than only $\\{Y_t\\}$ terms. Thus, more manipulations are required to build the GA-NoVaS method. Taking \\cref{4e1} as the starting point, we first find out expressions of $\\sigma_{t-1}^2,\\sigma_{t-2}^2,\\cdots$ as follow:\n\\begin{equation}\n \\begin{split}\n \\sigma_{t-1}^2 &= a + a_1Y_{t-2}^2 + b_1\\sigma_{t-2}^2\\\\\n \\sigma_{t-2}^2 &= a + a_1Y_{t-3}^2 + b_1\\sigma_{t-3}^2\\\\\n \\vdots& \\label{4e2}\n \\end{split}\n\\end{equation}\nPlug all components in \\cref{4e2} into \\cref{4e1}, one equation sequence can be gotten:\n\\begin{equation}\n \\begin{split}\n Y_t &= W_t\\sqrt{a + a_1Y_{t-1}^2 + b_1\\sigma_{t-1}^2}\\\\ \n &= W_t\\sqrt{a + a_1Y_{t-1}^2 + b_1(a + a_1Y_{t-2}^2 + b_1\\sigma_{t-2}^2)}\\\\\n &= W_t\\sqrt{a + a_1Y_{t-1}^2 + b_1a + b_1a_1Y_{t-2}^2 + b_1^2(a + a_1Y_{t-3}^2 + b_1\\sigma_{t-3}^2)}\\\\\n &\\vdots \\label{4e3}\n \\end{split}\n\\end{equation}\nIterating the process in \\cref{4e3}, with the requirement of $a_1+b_1<1$ for the stationarity, the limiting form of $Y_t$ can be written as \\cref{4e4}:\n\\begin{equation}\n Y_t =W_t\\sqrt{ \\sum_{i = 1}^{\\infty}a_1b_1^{i-1}Y_{t-i}^2 + \\sum_{j=0}^{\\infty}ab_1^j} = W_t\\sqrt{ \\sum_{i = 1}^{\\infty}a_1b_1^{i-1}Y_{t-i}^2 + \\frac{a}{1-b_1}} \\label{4e4}\n\\end{equation}\nWe can rewrite \\cref{4e4} to get a potential function $H_n$ which is corresponding to the GA-NoVaS method:\n\\begin{equation}\n W_t = \\frac{Y_t}{\\sqrt{ \\sum_{i = 1}^{\\infty}a_1b_1^{i-1}Y_{t-i}^2 + \\frac{a}{1-b_1}}} \\label{4e5}\n\\end{equation}\nRecall the adjustment taken in the existing GE-NoVaS method, the total difference between \\cref{3.2e2,3.2e3} can be seen as the term $a$ being replaced by $\\alpha s_{t-1}^2 + \\beta Y_t^2$. Apply this same adjustment on \\cref{4e5}, then this equation will be changed to the form as follows:\n\\begin{equation}\n W_t = \\frac{Y_t}{\\sqrt{ \\frac{\\beta Y_t^2 + \\alpha s_{t-1}^2}{1-b_1}+ \\sum_{i = 1}^{\\infty}a_1b_1^{i-1}Y_{t-i}^2 }} = \\frac{Y_t}{\\sqrt{ \\frac{\\beta Y_t^2}{1-b_1}+ \\frac{\\alpha s_{t-1}^2}{1-b_1} + \\sum_{i = 1}^{\\infty}a_1b_1^{i-1}Y_{t-i}^2 }} \\label{4e6}\n\\end{equation}\n In \\cref{4e6}, since $\\alpha\/(1-b_1)$ is also required to take a small positive value, this term can be seen as a $\\Tilde{\\alpha}$ ($\\Tilde{\\alpha} \\geq 0$) which is equivalent with $\\alpha$ in the existing GE-NoVaS method. Thus, we can simplify $\\alpha s_{t-1}^2\/(1-b_1)$ to $\\Tilde{\\alpha} s_{t-1}^2$. For keeping the same notation style with the GE-NoVaS method, we use $\\alpha s_{t-1}^2$ to represent $\\alpha s_{t-1}^2\/(1-b_1)$. Then \\cref{4e6} can be represented as:\n \\begin{equation}\n W_t = \\frac{Y_t}{\\sqrt{ \\frac{\\beta Y_t^2}{1-b_1}+ \\alpha s_{t-1}^2 + \\sum_{i = 1}^{\\infty}a_1b_1^{i-1}Y_{t-i}^2 }} \\label{4e7}\n\\end{equation}\n For getting a qualified GA-NoVaS transformation, we still need to make the transformation function \\cref{4e7} satisfy the requirement of the Model-free Prediction Principle. Recall that in the existing GE-NoVaS method, $\\alpha + \\beta + \\sum_{i=1}^pa_i$ in \\cref{3.2e3} is restricted to be 1 for meeting the requirement of variance-stabilizing and the optimal combination of $\\alpha,\\beta, a_1,\\cdots,a_p$ is selected to make the empirical distribution of $\\{W_t\\}$ as close to the standard normal distribution as possible (i.e., minimizing $\\abs{KURT(W_t)-3}$). Similarly, for getting a qualified $H_n$ from \\cref{4e7}, we require:\n \\begin{equation}\n \\frac{\\beta}{1-b_1} +\\alpha + \\sum_{i=1}^{\\infty}a_1b_1^{i-1} = 1 \\label{4e8}\n \\end{equation}\n Under this requirement, since $a_1$ and $b_1$ are both less than 1, $a_1b_1^{i-1}$ will converge to 0 as $i$ converges to $\\infty$, i.e., $a_1b_1^{i-1}$ is neglectable when $i$ takes large values. So it is reasonable to replace $\\sum_{i=1}^{\\infty}a_1b_1^{i-1}$ in \\cref{4e8} by $\\sum_{i=1}^{q}a_1b_1^{i-1}$, where $q$ takes a large value. Then a truncated form of \\cref{4e7} can be written as \\cref{4e9}:\n\\begin{equation}\n W_t = \\frac{Y_t}{\\sqrt{ \\frac{\\beta Y_t^2}{1-b_1}+ \\alpha s_{t-1}^2 + \\sum_{i = 1}^{q}a_1b_1^{i-1}Y_{t-i}^2 }}~;~\\text{for}~ t=q+1,\\cdots,n. \\label{4e9}\n\\end{equation}\nNow, we take \\cref{4e9} as a potential function $H_n$. Then, the requirement of variance-stabilizing is changed to:\n\\begin{equation}\n \\frac{\\beta}{1-b_1} +\\alpha + \\sum_{i=1}^{q}a_1b_1^{i-1} = 1\\label{4e10}\n\\end{equation}\n\\\\\nAkin to \\cref{3.2e6}, we scale $\\{\\frac{\\beta}{1-b_1},a_1,a_1b_1$ $,a_1b_1^{2},$ $\\cdots,a_1b_1^{q-1} \\}$ of \\cref{4e10} by timing a scalar $\\frac{1-\\alpha}{\\frac{\\beta}{1-b_1} + \\sum_{i=1}^{q}a_1b_1^{i-1}}$, and then search optimal coefficients. For presenting \\cref{4e9} with scaling coefficients in a concise form, we use $\\{c_0,c_1,\\cdots,c_q\\}$ to represent $\\{\\frac{\\beta}{1-b_1},a_1,a_1b_1$ $,a_1b_1^{2},$ $\\cdots,a_1b_1^{q-1} \\}$ after scaling, which implies that we can rewrite \\cref{4e9} as:\n\\begin{equation}\n W_t = \\frac{Y_t}{\\sqrt{ c_0Y_t^2+ \\alpha s_{t-1}^2 + \\sum_{i = 1}^{q}c_iY_{t-i}^2 }}~;~\\text{for}~ t=q+1,\\cdots,n. \\label{4e9v}\n\\end{equation}\n\\begin{remark}(The difference between GA-NoVaS and GE-NoVaS methods)\nCompared with the existing GE-NoVaS method, we should notice that the GA-NoVaS method possesses a totally different transformation structure. Recall all coefficients except $\\alpha$ implied by the GE-NoVaS method are expressed as $\\beta = c', a_i = c'e^{-ci}~$ $\\text{for all}~1\\leq$ $i\\leq p$, $c' = \\frac{1-\\alpha}{\\sum_{j=0}^pe^{-cj}}$. There are only two free parameters $c$ and $\\alpha$. However, there are four free parameters $\\beta, a_1, b_1$ and $\\alpha$ in \\cref{4e9}. For example, the coefficient of $Y_t^2$ of the GE-NoVaS method is $(1-\\alpha)\/(\\sum_{j=0}^pe^{-cj})$. On the other hand, the corresponding coefficient in the GA-NoVaS structure is $c_0 = \\beta(1-\\alpha)\/(\\beta+(1-b_1)\\sum_{i=1}^{q}a_1b_1^{i-1})$. We can think the freedom of coefficients within the GA-NoVaS is larger than the freedom in the GE-NoVaS. At the same time, the structure of GA-NoVaS method is built from GARCH(1,1) model directly without imposing any prior assumption on coefficients. We believe this is the reason why our GA-NoVaS method shows better prediction performance in \\cref{sec:simu,sec:real data}. \n\\end{remark}\n\nFurthermore, for achieving the aim of normalizing, we still fix $\\alpha$ to be one specific value from $\\{0.1,0.2,\\cdots,0.8\\}$, and then search the optimal combination of $\\beta,a_1,b_1$ from three grids of possible values of $\\beta,a_1,b_1$ to minimize $\\abs{KURT(W_t)-3}$. After getting a qualified $H_n$, $H_n^{-1}$ will be outlined immediately:\n\\begin{equation}\n Y_t = \\sqrt{\\frac{W_t^2}{1-c_0W_t^2}(\\alpha s_{t-1}^2+\\sum_{i=1}^qc_iY_{t-i}^2)}~;~\\text{for}~ t=q+1,\\cdots,n. \\label{4e11}\n\\end{equation}\nBased on \\cref{4e11}, $Y_{n+1}$ can be expressed as the equation follows:\n\\begin{equation}\n Y_{n+1} = \\sqrt{\\frac{W_{n+1}^2}{1-c_0W_{n+1}^2}(\\alpha s_{n}^2+\\sum_{i=1}^qc_iY_{n+1-i}^2)} \\label{4e12}\n\\end{equation}\nAlso, it is not hard to express $Y_{n+h}$ as a function of $W_{n+1},\\cdots, W_{n+h} $ and $\\mathscr{F}_{n}$ with GA-NoVaS method like we did in \\cref{ssec:genovas}:\n\\begin{equation}\n Y_{n+h} = f_{GA}(W_{n+1},\\cdots,W_{n+h};\\mathscr{F}_{n})~;~\\text{for any}~h\\geq 1. \\label{4e13}\n\\end{equation}\n\\par\n\\noindent Once the expression of $Y_{n+h}$ is figured out, we can apply the same procedure with the GE-NoVaS method to get the optimal predictor of $Y_{n+h}$ under $L_1$ or $L_2$ risk criterion. To deal with $\\alpha$, we still adopt the same strategy used in the GE-NoVaS method, i.e., select the optimal $\\alpha$ from a grid of possible values based on prediction performance. One thing should be noticed is that the value of $\\alpha$ is invariant during the process of optimization once we fix it as a specific value. More details about the algorithm of this new method can be found in \\cref{ssc:algorithm}.\n\n\\subsection{Parsimonious variant of the GA-NoVaS method}\\label{subsec:parsimoniousvariant}\nAccording to the $\\beta$-removing idea, we can continue proposing the GA-NoVaS-without-$\\beta$ method which is a parsimonious variant of the GA-NoVaS method. From \\cite{wu2021boosting}, functions $H_n$ and $H_n^{-1}$ corresponding to the GE-NoVaS-without-$\\beta$ method can be presented as follow:\n\\begin{equation}\n W_{t}=\\frac{Y_t}{\\sqrt{\\alpha s_{t-1}^2+\\sum_{i=1}^pa_iY_{t-i}^2}}~;~Y_t=\\sqrt{W_{t}^2(\\alpha s_{t-1}^2+\\sum_{i=1}^pa_iY_{t-i}^2)}~;~\\text{for}~ t=p+1,\\cdots,n. \\label{eq:3e15} \n\\end{equation}\n\\cref{eq:3e15} still need to satisfy the requirement of normalizing and variance-stabilizing transformation. Therefore, we restrict $\\alpha + \\sum_{i=1}^pa_i = 1$ and still select the optimal combination of $ a_1,\\cdots,a_p$ by minimizing $\\abs{KURT(W_t)-3}$. Then, $Y_{n+1}$ can be expressed by \\cref{eq:3e16}:\n\\begin{equation}\n Y_{n+1}=\\sqrt{W_{n+1}^2(\\alpha s_{n}^2+\\sum_{i=1}^pa_iY_{n+1-i}^2)} \\label{eq:3e16}\n\\end{equation}\n\\begin{remark}Even though we do not include the effect of $Y_t$ when we build $H_n$, the expression of $Y_{n+1}$ still contains the current value $Y_n$. It means the GE-NoVaS-without-$\\beta$ method does not disobey the rule of causal prediction.\n\\end{remark}\n\nSimilarly, our proposed GA-NoVaS method can also be offered in a variant without $\\beta$ term. \\cref{4e9v,4e11} without $\\beta$ term can be represented by following equations:\n\\begin{equation}\n W_t = \\frac{Y_t}{\\sqrt{ \\alpha s_{t-1}^2 + \\sum_{i = 1}^{q}\\Tilde{c}_iY_{t-i}^2 }}~;~Y_t = \\sqrt{W_t^2(\\alpha s_{t-1}^2+\\sum_{i=1}^q\\Tilde{c}_iY_{t-i}^2)} \\label{4e21}\n\\end{equation}\nOne thing should be mentioned here is that $\\{\\Tilde{c}_1,\\cdots,\\Tilde{c}_q\\}$ represents $\\{a_1,a_1b_1$ $,a_1b_1^{2},$ $\\cdots,a_1b_1^{q-1} \\}$ scaled by timing a scalar $\\frac{1-\\alpha}{\\sum_{j=1}^{q}a_1b_1^{j-1}}$. Besides, $\\alpha + \\sum_{i=1}^{q}\\Tilde{c}_i = 1$ is required to satisfy the variance-stabilizing requirement and the optimal combination of $a_1,b_1$ is selected by minimizing $\\abs{KURT(W_t)-3}$ to satisfy the normalizing requirement. For GE-NoVaS- and GA-NoVaS-without-$\\beta$ methods, we can still express $Y_{n+h}$ as a function of $\\{W_{n+1},\\cdots,W_{n+h}\\}$ and repeat the aforementioned procedure to get $L_1$ and $L_2$ predictors. For example, we can derive the expression of $Y_{n+h}$ using the GA-NoVaS-without-$\\beta$ method:\n\n\\begin{equation}\n Y_{n+h} = f_{\\text{GA-without-$\\beta$}}(W_{n+1},\\cdots,W_{n+h};\\mathscr{F}_{n})~;~\\text{for any}~h\\geq 1. \\label{4e22}\n\\end{equation}\n\n\\begin{remark}[Slight computational efficiency of removing $\\beta$]\\label{remark3.2}\nNote that the suggestion of removing $\\beta$ can also lead a less time-complexity of the existing GE-NoVaS and newly proposed GA-NoVaS methods. The reason for this is simple: Recall $1\/\\sqrt{\\beta}$ is required to be larger or equal to 3 for making $\\{W_t\\}$ have enough large range, i.e., $\\beta$ is required to be less or equal to 0.111. However, the optimal combination of NoVaS coefficients may not render a suitable $\\beta$. For this situation, we need to increase the time-series order ($p$ or $q$) and repeat the normalizing and variance-stabilizing process till $\\beta$ in the optimal combination of coefficients is appropriate. This replication process definitely increases the computation workload.\n\\end{remark}\n\n\\subsection{Connection of two parsimonious methods}\\label{ssec:connection}\nIn this subsection, we reveal that GE-NoVaS-without-$\\beta$ and GA-NoVaS-without-$\\beta$ methods actually have a same structure. The difference between these two methods lies in the region of free parameters. For observing this phenomenon, let us consider scaled coefficients of GA-NoVaS-without-$\\beta$ method except $\\alpha$:\n\\begin{equation}\n\\left\\{ \\frac{(1-\\alpha)b_1^{i-1}}{\\sum_{j=1}^{q}b_1^{j-1}}\\right\\}_{i=1}^{q} =\\left\\{ \\frac{(1-\\alpha)b_1^{i}}{\\sum_{j=1}^{q}b_1^{j}}\\right\\}_{i=1}^{q} \\label{eq:3.19}\n\\end{equation}\nRecall parameters of GE-NoVaS-without-$\\beta$ method except $\\alpha$ implied by \\cref{3.2e6} are:\n\\begin{equation}\n \\left\\{ \\frac{(1-\\alpha)e^{-ci}}{\\sum_{j=1}^pe^{-cj}} \\right\\}_{i=1}^{p} \\label{eq:3.20}\n\\end{equation}\n\nObserving above two equations, although we can discover that \\cref{eq:3.19} and \\cref{eq:3.20} are equivalent if we set $b_1$ being equal to $e^{-c}$, these two methods are still slightly different since regions of $b_1$ and $c$ play a role in the process of optimization. The complete region of $c$ could be $(0,\\infty)$. However, \\cite{politis2015modelfreepredictionprinciple} pointed out that $c$ can not take a large value\\footnote{When $c$ is large, $a_i \\approx 0$ for all $i > 0$. It is hard to make the kurtosis of transformed series be 3.} and the region of $c$ should be an interval of the type $(0,m)$ for some $m$. In other words, a formidable search problem for finding the optimal $c$ is avoided by choosing such trimmed interval. On the other hand, $b_1$ is explicitly searched from $(0,1)$ which is corresponding with $c$ taking values from $(0,\\infty)$. Likewise, applying the GA-NoVaS-without-$\\beta$ method, the aforementioned burdensome search problem is also eliminated. Moreover, we can build a transformation based on the whole available region of unknown parameter. In spite of the fact that GE-NoVaS-without-$\\beta$ and GA-NoVaS-without-$\\beta$ methods have indistinguishable prediction performance for most of data analysis cases, we argue that the GA-NoVaS-without-$\\beta$ method is more stable and reasonable than the GE-NoVaS-without-$\\beta$ method since it is a more complete technique viewing the available region of parameter. Moreover, GA-NoVaS-without-$\\beta$ method achieves significantly better prediction performance for some cases, see more details from \\hyperref[appendix:a]{Appendix A}.\n\n\\subsection{Algorithms of new methods}\\label{ssc:algorithm}\n\\noindent In \\cref{ssc:garchnovas,subsec:parsimoniousvariant}, we exhibit the GA-NoVaS method and its parsimonious variant. In this section, we provide algorithms of these two methods. For the GA-NoVaS method, unknown parameters $\\beta, a_1, b_1$ are selected from three grids of possible values to normalize $\\{W_t;~t = q+1,\\cdots,n\\}$ in \\cref{4e9}. If our goal is the $h$-step ahead prediction of $g(Y_{n+h})$ using past $\\{Y_t;~t=1,\\cdots,n\\}$, the algorithm of the GA-NoVaS method can be summarized in \\cref{algori1}.\n\n\\begin{algorithm}[htbp]\n\\caption{the $h$-step ahead prediction for the GA-NoVaS method}\n\\centering\n\\label{algori1}\n \\centering\n \\begin{tabular} {p{29pt}p{280pt}} \n Step 1 & Define a grid of possible $\\alpha$ values, $\\{\\alpha_k;~ k = 1,\\cdots,K\\}$, three grids of possible $\\beta$, $a_1$, $b_1$ values. Fix $\\alpha = \\alpha_k$, then calculate the optimal combination of $\\beta,a_1,b_1$ of the GA-NoVaS method.\\\\\n Step 2 & Derive the analytic form of \\cref{4e13} using $\\{\\beta, a_1, b_1, \\alpha_k\\}$ from the first step.\\\\\n Step 3 & Generate $\\{W_{n+1,m},\\cdots, W_{n+h,m}\\}_{m=1}^{M}$ from a trimmed standard normal distribution or empirical distribution $\\hat{F}_w$. Plug $\\{W_{n+1,m},\\cdots, W_{n+h,m}\\}_{m=1}^{M}$ into the analytic form of \\cref{4e13} to obtain $M$ pseudo-values $\\{Y_{n+h,m}\\}_{m=1}^{M}$.\\\\\n Step 4 & Calculate the optimal predictor $g(\\hat{Y}_{n+h})$ by taking the sample mean (under $L_2$ risk criterion) or sample median (under $L_1$ risk criterion) of the set $\\{g(Y_{n+h,1}),\\cdots,g(Y_{n+h,M})\\}$.\\\\\n Step 5 & Repeat above steps with different $\\alpha$ values from $\\{\\alpha_k;~ k = 1,\\cdots,K\\}$ to get $K$ prediction results. \\\\ \n \\end{tabular}\n\\end{algorithm}\nIf we want to apply the GA-NoVaS-without-$\\beta$ method, we just need to change \\cref{algori1} a little bit. The difference between \\cref{algori1,algori2} is the optimization of $\\beta$ term being removed. The optimal combination of $a_1,b_1$ is still selected based on the normalizing and variance-stabilizing purpose. In our experiment setting, we choose regions of $\\beta,a_1,b_1$ being $(0,1)$ and set a 0.02 grid interval to find all parameters. Besides, for the GA-NoVaS method, we also make sure that the sum of $\\beta,a_1,b_1$ is less than 1 and the coefficient of $Y_t^{2}$ is the largest one. \\\\ \n\\begin{algorithm}[H]\n\\centering\n\\caption{the $h$-step ahead prediction for the GA-NoVaS-without-$\\beta$}\n\\label{algori2}\n\\hspace{0.5cm}\n \\centering\n \\begin{tabular} {p{29pt}p{280pt}} \n \\centering Step 1 & Define a grid of possible $\\alpha$ values, $\\{\\alpha_k;~ k = 1,\\cdots,K\\}$, two grids of possible $a_1$, $b_1$ values. Fix $\\alpha = \\alpha_k$, then calculate the optimal combination of $a_1,b_1$ of the GA-NoVaS-without-$\\beta$ method.\\\\\n \\centering Steps 2-5 & Same as \\cref{algori1}, but $\\{W_{n+1,m},\\cdots, W_{n+h,m}\\}_{m=1}^{M}$ are plugged into the analytic form of \\cref{4e22} and the standard normal distribution does not need to be truncated.\n \\\\\n \\end{tabular}\n\\end{algorithm}\n\n\n\\section{Simulation}\\label{sec:simu}\n\n\\noindent In simulation studies, for controlling the dependence of prediction performance on the length of the dataset, 16 datasets (2 from each settings) are generated from different GARCH(1,1)-type models separately and the size of each dataset is 250 (short data mimics 1-year of econometric data) or 500 (large data mimics 2-years of econometric data).\n\\\\\n\\\\\n\\textbf{Model 1:} Time-varying GARCH(1,1) with Gaussian errors\\\\\n$X_t = \\sigma_t\\epsilon_t,~\\sigma_t^2 = \\omega_{0,t} + \\beta_{1,t}\\sigma_{t-1}^2+\\alpha_{1,t}X_{t-1}^2,~\\{\\epsilon_t\\}\\sim i.i.d.~N(0,1)$\\\\\n$g_t = t\/n; \\omega_{0,t}= -4sin(0.5\\pi g_t)+5; \\alpha_{1,t} = -1(g_t-0.3)^2 + 0.5; \\beta_{1,t} = 0.2sin(0.5\\pi g_t)+0.2,~n = 250~\\text{or}~500$\\\\\n\\textbf{Model 2:} Another time-varying GARCH(1,1) with Gaussian errors\\\\\n$X_t = \\sigma_t\\epsilon_t,~\\sigma_t^2 = 0.00001 + \\beta_{1,t}\\sigma_{t-1}^2+\\alpha_{1,t}X_{t-1}^2,~\\{\\epsilon_t\\}\\sim i.i.d.~N(0,1)$\\\\\n$g_t = t\/n$; $\\alpha_{1,t} = 0.1 - 0.05g_t$; $\\beta_{1,t} = 0.73 + 0.2g_t,~n = 250~\\text{or}~500$\\\\\n\\textbf{Model 3:} Standard GARCH(1,1) with Gaussian errors\\\\\n$X_t = \\sigma_t\\epsilon_t,~\\sigma_t^2 = 0.00001 + 0.73\\sigma_{t-1}^2+0.1X_{t-1}^2,~\\{\\epsilon_t\\}\\sim i.i.d.~N(0,1)$\\\\\n\\textbf{Model 4:} Standard GARCH(1,1) with Gaussian errors\\\\\n$X_t = \\sigma_t\\epsilon_t,~\\sigma_t^2 = 0.00001 + 0.8895\\sigma_{t-1}^2+0.1X_{t-1}^2,~\\{\\epsilon_t\\}\\sim i.i.d.~N(0,1)$\\\\\n\\textbf{Model 5:} Standard GARCH(1,1) with Student-$t$ errors\\\\\n$X_t = \\sigma_t\\epsilon_t,$ $~\\sigma_t^2 = 0.00001 + 0.73\\sigma_{t-1}^2+0.1X_{t-1}^2,$\\\\ $~\\{\\epsilon_t\\}\\sim i.i.d.~t$ $\\text{distribution with five degrees of freedom}$\\\\\n\\textbf{Model 6:} Exponential GARCH(1,1) with Gaussian errors\\\\\n$X_t = \\sigma_t\\epsilon_t,~\\log(\\sigma_t^2) = 0.00001 + 0.8895\\log(\\sigma^2_{t-1})+0.1\\epsilon_{t-1}+0.3(\\abs{\\epsilon_{t-1}}-E\\abs{\\epsilon_{t-1}}),$\\\\$~\\{\\epsilon_t\\}\\sim i.i.d.~N(0,1)$\\\\\n\\textbf{Model 7:} GJR-GARCH(1,1) with Gaussian errors\\\\\n$X_t = \\sigma_t\\epsilon_t,~\\sigma_t^2 = 0.00001 + 0.5\\sigma^2_{t-1}+0.5X_{t-1}^2-0.5I_{t-1}X_{t-1}^2,~\\{\\epsilon_t\\}\\sim i.i.d.~N(0,1)\\\\\nI_{t} = 1~\\text{if}~ X_t \\leq 0; I_{t} = 0~ \\text{otherwise}$\\\\\n\\textbf{Model 8:} Another GJR-GARCH(1,1) with Gaussian errors\\\\\n$X_t = \\sigma_t\\epsilon_t,~\\sigma_t^2 = 0.00001 + 0.73\\sigma^2_{t-1}+0.1X_{t-1}^2+0.3I_{t-1}X_{t-1}^2,~\\{\\epsilon_t\\}\\sim i.i.d.~N(0,1)\\\\\nI_{t} = 1~\\text{if}~ X_t \\leq 0; I_{t} = 0~ \\text{otherwise}$\\\\\n\n\\textit{Model description:} Models 1 and 2 present a time-varying GARCH model where coefficients $a_0, a_1, b_1$ change over time slowly. They differ significantly in the intercept term of $\\sigma_t^2$ as we intentionally keep it low in the second setting. Models 3 and 4 are from a standard GARCH where in Model 4 we wanted to explore a scenario that $\\alpha_1+\\beta_1$ is very close to 1 and thus mimic what would happen for the iGARCH situation. Model 5 allows for the error distribution to come from a student-$t$ distribution instead of the Gaussian distribution. Note that, for a fair competition, we chose Models 2 to 5 same as simulation settings of \\citep{chen2019optimal}. Models 6, 7 and 8 present different types of GARCH models. These settings allow us to check robustness of our methods against model misspecification. In a real world, it is hard to convincingly say if the data obeys one particular type of GARCH model, so we want to pursue this exercise to see if our methods are satisfactory no matter what the underlying distribution and the GARCH-type model are. This approach to test the performance of a method under model misspecification is quite standard, see \\cite{olubusoye2016misspecification} used data generated from a specifically true model to estimate other GARCH-type models and test the forecasting performance, and \\cite{bellini2008misspecification} investigated the impact of misspecification of innovations in fitting GARCH models.\n\n\\textit{Window size:} Using these datasets, we perform 1-step, 5-steps and 30-steps ahead time-aggregated POOS predictions. For measuring different methods' prediction performance on larger datasets (i.e., data size is 500), we use 250 data as a window to do predictions and roll this window through the whole dataset. For evaluating different methods' performance on smaller datasets (i.e., data size is 250), we use 100 data as a window. \n\nNote that log-returns can be calculated from equation shown below:\n\\begin{equation}\n Y_t = 100\\times \\log(X_{t+1}\/X_t) ~;~\\text{for}~ t = 1,\\cdots,499~\\text{or}~t = 1,\\cdots,249. \\label{Eq:4.1}\n\\end{equation}\nWhere, $\\{X_t\\}_{t = 1}^{250}$ and $\\{X_t\\}_{t = 1}^{500}$ are 1-year and 2-years price series, respectively. Next, we can define time-aggregated predictions of squared log-returns as:\n\\begin{equation}\n\\begin{split}\n \\bar{Y}_{k,1}^2 = \\hat{Y}_{k+1}^2,~k=250,\\cdots,498 ~\\text{or}~k=100,\\cdots,248\\\\\n \\bar{Y}_{i,5}^2 = \\frac{1}{5}\\sum_{m=1}^5\\hat{Y}^2_{i+m},~i = 250,\\cdots,494~\\text{or}~i=100,\\cdots,244\\\\\n \\bar{Y}_{j,30}^2 = \\frac{1}{30}\\sum_{m=1}^{30}\\hat{Y}^2_{j+m},~j = 250,\\cdots,469~\\text{or}~j=100,\\cdots,219 \\label{4e17}\n\\end{split}\n\\end{equation}\nIn \\cref{4e17}, $\\hat{Y}_{k+1}^2,\\hat{Y}_{i+m}^2,\\hat{Y}_{j+m}^2$ are single point predictions of realized squared log-returns by NoVaS-type methods or benchmark method; $\\bar{Y}_{k,1}^2$, $\\bar{Y}_{i,5}^2$ and $\\bar{Y}_{j,30}^2$ represent 1-step, 5-steps and 30-steps ahead aggregated predictions, respectively. More specifically, for exploring the performance of three different prediction lengths with large data size, we roll the 250 data points window through the whole dataset, i.e., use $\\{Y_1,\\cdots,Y_{250}\\}$ to predict $Y_{251}^2,\\{Y_{251}^2,\\cdots,Y_{255}^2\\}$ and $\\{Y_{251}^2,\\cdots,Y_{280}^2\\}$; then use $\\{Y_2,\\cdots,Y_{251}\\}$ to predict $Y_{252}^2,\\{Y_{252}^2,\\cdots,Y_{256}^2\\}$ and $\\{Y_{252}^2,\\cdots,Y_{281}^2\\}$, for 1-step, 5-steps and 30-steps aggregated predictions respectively, and so on. For exploring the performance of three different prediction lengths with small data size, we roll the 100 data points window through the whole dataset, i.e., use $\\{Y_1,\\cdots,Y_{100}\\}$ to predict $Y_{101}^2,\\{Y_{101}^2,\\cdots,Y_{105}^2\\}$ and $\\{Y_{101}^2,\\cdots,Y_{130}^2\\}$; then use $\\{Y_2,\\cdots,Y_{101}\\}$ to predict $Y_{102}^2,\\{Y_{102}^2,\\cdots,Y_{106}^2\\}$ and $\\{Y_{102}^2,\\cdots,Y_{131}^2\\}$, for 1-step, 5-steps and 30-steps aggregated predictions respectively, and so on. For example, with window size being 30, we perform time-aggregated predictions on a large dataset 220 times. Taking this strategy, we can exhaust the information contained in the dataset and investigate the forecasting performance continuously. \n\nTo measure different methods' forecasting performance, we compare predictions with realized values based on \\cref{eq:4.1}. \n\\begin{equation}\n P = \\sum_{l}(\\bar{Y}_{l,h}^2-\\sum_{m=1}^h(Y_{l+m}^2\/h))^2~;~l \\in \\{k,i,j\\}\\label{eq:4.1}\n\\end{equation}\nIn \\cref{eq:4.1}, setting $l = k,i,j$ means we consider 1-step, 5-steps and 30-steps ahead time-aggregated predictions respectively; $\\bar{Y}_{l,h}^2$ is the $h$-step ($h\\in\\{1,5,30\\}$) ahead time-aggregated volatility prediction, defined in \\cref{4e17}; $\\sum_{m=1}^h(Y_{l+m}^2\/h)$ is the corresponding true aggregated value calculated from realized squared log-returns. For comparing various Model-free methods with the traditional method, we set the benchmark method as fitting one GARCH(1,1) model directly (GARCH-direct).\n\n\\textit{Different variants of methods:} Note that we can perform GE-NoVaS-type and GA-NoVaS-type methods to predict $Y_{n+h}$ by generating $\\{W_{n+1,m},\\cdots, W_{n+h,m}\\}_{m=1}^{M}$ from a standard normal distribution or the empirical distribution of $\\{W_t\\}$ series, then we can calculate the optimal predictor based on $L_1$ or $L_2$ risk criterion. It means each NoVaS-type method possesses four variants. \n\nWhen we perform POOS forecasting, we do not know which $\\alpha$ is optimal. Thus, we perform every NoVaS variants using $\\alpha$ from eight potential values $\\{0.1, 0.2, \\cdots,0.8\\}$ and then pick the optimal result. For simplifying the presentation, we further select the final prediction from optimal results of four variants of a NoVaS method and use this result to be the best prediction to which each NoVaS method can reach. Applying this procedure means we take a computationally heavy approach to compare different methods' potentially best performance. However, it also means we want to challenge newly proposed methods at a maximum level, so as to see if they can beat even the best-performing scenario of the current GE-NoVaS method. \n\n\\subsection{Simulation results}\\label{ssec:simuresults}\n\\noindent In this subsection, we compare the performance of our new methods (GA-NoVaS and GA-NoVaS-without-$\\beta$) with GARCH-direct and existing GE-NoVaS methods on forecasting 250 and 500 simulated data. Results are tabulated in \\cref{5t1}.\n\n\\subsubsection{Simulation results of Models 1 to 5}\\label{sssec:simuresultsmoeld1-5}\n\\noindent From \\cref{5t1}, we clearly find NoVaS-type methods outperform the GARCH-direct method. Especially for using the 500 Model-1 data to do 30-steps ahead aggregated prediction, the performance of the GARCH-direct method is terrible. NoVaS-type methods are almost 30 times better than the GARCH-direct method. This means that the normal prediction method may be spoiled by error accumulation problem when long-term predictions are required. On the other hand, Model-free methods can avoid this problem.\n\nIn addition to the overall advantage of NoVaS-type methods over GARCH-direct method, we find the GA-NoVaS method is generally better than the GE-NoVaS method for both short and large data. This conclusion is two-fold: (1) The time of the GA-NoVaS being the best method is more than the GE-NovaS method; (2) Since we want to compare the forecasting ability of GE-NoVaS and GA-NoVaS methods, we use $*$ symbol to represent cases where the GA-NoVaS method works at least 10$\\%$ better than the GE-NoVaS method or inversely the GE-NoVaS method is 10$\\%$ better. We can find there is no case to support that the GE-NoVaS works better than GA-NoVaS with as least 10$\\%$ improvement. On the other hand, the GA-NoVaS method achieves significant improvement when long-term predictions are required. Moreover, the GA-NoVaS-without-$\\beta$ dominates other two NoVaS-type methods.\n\n\\subsubsection{Models 6 to 8: Different GARCH specifications}\\label{sssc:simudifferent}\n\\noindent Since the main crux of Model-free methods is how such non-parametric methods are robust to underlying data-generation processes, here we explore other GARCH-type data generations. The GA-NoVaS method is based upon GARCH model, so it is interesting to explore whether even these methods can sustain a different type of true underlying generation and can in general outperform existing methods. Results for Models 6 to 8 are tabulated in \\cref{5t1}.\n\nIn general, NoVaS-type methods still outperform the GARCH-direct method for these cases. Although the forecasting ability of GE-NoVaS and GA-NoVaS for large data is indistinguishable, the GA-NoVaS is obviously better for taking short data size. For example, the GA-NoVaS method brings around 20$\\%$ improvement compared with the GE-NoVaS method for 30-steps ahead aggregated prediction of 250 Model-6 simulated data. Doing better prediction with past data that is shorter in size is always a significant challenge and thus it is valuable to discover the GA-NoVaS method has superior performance for this scenario. Not surprisingly, the GA-NoVaS-without-$\\beta$ method still keeps great performance.\n\n\\subsection{Simulation summary}\\label{ssc:simusmall}\n\\noindent Through deploying simulation data analysis, we find GA-NoVaS-type methods can sustain great performance against short data and model misspecification. Overall, our new methods outperform the GE-NoVaS method and can render notable improvement for some cases when long-term predictions are desired. \n\n\\begin{table}[htbp]\n\\caption{Comparison results of using 500 and 250 simulated data}\n\\label{5t1}\n\\begin{adjustbox}{width=1\\textwidth}\n\\small\n\\begin{tabular}{lcccclcccc}\n \\toprule\n \\textbf{500} size & \\thead{\\small GE} & \\thead{\\small GA} & \\thead{\\small P-GA} & \\thead{\\small GARCH} & \\textbf{250} size & \\thead{\\small GE} & \\thead{\\small GA} & \\thead{\\small P-GA} & \\thead{\\small GARCH}\\\\ \n \\midrule\n\n M1-1step & 0.89258 & 0.88735 & \\textbf{0.84138} & 1.00000 & M1-1step & 0.91538 & 0.9112 & \\textbf{0.83034} & 1.00000 \\\\ [2pt]\n M1-5steps & 0.40603 & 0.40296 & \\textbf{0.40137} & 1.00000 & M1-5steps & 0.49169 & 0.48479 & \\textbf{0.43247} & 1.00000 \\\\[2pt]\n M1-30steps & 0.03368 & 0.03294 & \\textbf{0.03290} & 1.00000 & M1-30steps & 0.25009 & 0.24752 & \\textbf{0.23035} & 1.00000 \\\\[2pt]\n M2-1step & \\textbf{0.95689} & 0.96069 & 0.99658 & 1.00000 & M2-1step & 0.91369 & 0.91574 & \\textbf{0.87614} & 1.00000 \\\\[2pt]\n M2-5steps & 0.89981 & \\textbf{0.89739} & 0.9198 & 1.00000 & M2-5steps & 0.61001 & 0.61094 & \\textbf{0.51712} & 1.00000 \\\\[2pt]\n M2-30steps & 0.63126 & 0.64042 & \\textbf{0.48396} & 1.00000 & M2-30steps & 0.7725 & \\textbf{0.74083} & 0.75251 & 1.00000 \\\\[2pt]\n M3-1step & 0.99938 & 1.00150 & \\textbf{0.98407} & 1.00000 & M3-1step & 0.97796 & 0.96632 & \\textbf{0.93693} & 1.00000 \\\\[2pt]\n M3-5steps & 0.98206 & 0.96088 & \\textbf{0.94073} & 1.00000 & M3-5steps & 0.98127 & \\textbf{0.97897} & 0.99977 & 1.00000 \\\\[2pt]\n M3-30steps & 1.10509 & 1.03683 & \\textbf{0.90855} & 1.00000 & M3-30steps & 1.38353 & \\textbf{0.89001*} & 0.99818 & 1.00000 \\\\[2pt]\n M4-1step & 0.98713 & \\textbf{0.98466} & 0.9964 & 1.00000 & M4-1step & 0.99183 & 0.95698 & \\textbf{0.92811} & 1.00000 \\\\[2pt]\n M4-5steps & 0.95382 & 0.95362 & \\textbf{0.95338} & 1.00000 & M4-5steps & 0.77088 & 0.72882 & \\textbf{0.67894} & 1.00000 \\\\[2pt]\n M4-30steps & 0.75811 & 0.69208 & \\textbf{0.67594} & 1.00000 & M4-30steps & 0.79672 & \\textbf{0.6095*} & 0.81115 & 1.00000 \\\\[2pt]\n M5-1step & 0.96940 & \\textbf{0.94066} & 0.97151 & 1.00000 & M5-1step & 0.83631 & 0.84134 & 0. \\textbf{79075} & 1.00000 \\\\[2pt]\n M5-5steps & 0.84751 & \\textbf{0.72806*} & 0.82747 & 1.00000 & M5-5steps & 0.38296 & 0.38034 & \\textbf{0.35155} & 1.00000 \\\\[2pt]\n M5-30steps & 0.49669 &\\textbf{ 0.24318*} & 0.47311 & 1.00000 & M5-30steps & 0.00199 & 0.002 & \\textbf{0.00194} & 1.00000 \\\\[2pt]\n M6-1step & 1.00175 & 1.00514 & \\textbf{0.93509} & 1.00000 & M6-1step & 0.95939 & 0.96499 & \\textbf{0.93863} & 1.00000 \\\\[2pt]\n M6-5steps & 0.93796 & 0.94249 & \\textbf{ 0.80311} & 1.00000 & M6-5steps & 0.93594 & 0.97101 & \\textbf{0.85851} & 1.00000 \\\\[2pt]\n M6-30steps & 0.50740 & 0.51350 & \\textbf{0.41112} & 1.00000 & M6-30steps & 0.84401 & \\textbf{0.67272*} & 0.7042 & 1.00000 \\\\[2pt]\n M7-1step & 0.98857 & 0.98737 & \\textbf{0.95932} & 1.00000 & M7-1step & 0.84813 & 0.83628 & \\textbf{0.83216} & 1.00000 \\\\[2pt]\n M7-5steps & 0.85539 & 0.85371 & \\textbf{0.85127} & 1.00000 & M7-5steps & 0.50849 & 0.50126 & \\textbf{0.4802} & 1.00000 \\\\[2pt]\n M7-30steps & \\textbf{0.68202} & 0.68314 & 0.71391 & 1.00000 & M7-30steps & 0.06832 & 0.06817 & \\textbf{0.06507} & 1.00000 \\\\[2pt]\n M8-1step & 0.96001 & 0.96463 & \\textbf{0.93452} & 1.00000 & M8-1step & \\textbf{0.79561} & 0.79994 & 0.8334 & 1.00000 \\\\[2pt]\n M8-5steps & 0.97019 & 0.98184 & \\textbf{0.93178} & 1.00000 & M8-5steps & 0.48028 & 0.47244 & \\textbf{0.45665} & 1.00000 \\\\[2pt]\n M8-30steps & \\textbf{0.30593} & 0.31813 & 0.33853 & 1.00000 & M8-30steps & 0.00977 & \\textbf{0.00942} & 0.00983 & 1.00000 \\\\[2pt]\n \\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\\\\n\\tiny \\textit{Note:} Column names ``GA'' and ``GE'' represent GE-NoVaS and GA-NoVaS methods, respectively; ``GARCH'' means GARCH-direct method; ``P-GA'' means GA-NoVaS-without-$\\beta$ method. The benchmark is the GARCH-direct method, so numerical values in the table corresponding to GARCH-direct method are 1. Other numerical values are relative values compared to the GARCH-direct method. ``$Mi\\text{-}j$''steps denotes using data generated from the Model $i$ to do $j$ steps ahead time-aggregated predictions. The bold value means that the corresponding method is the optimal choice for this data case. Cell with $*$ means the GA-NoVaS method is at least 10$\\%$ better than the GE-NoVaS method or inversely the GE-NoVaS method is at least 10$\\%$ better.\n\\end{table} \n\n\n\\section{Real-world data analysis}\\label{sec:real data}\n\\noindent From \\cref{sec:simu}, we have found that NoVaS-type methods have great performance on dealing with different simulated datasets. However, no methodological proposal is complete unless one verifies it on several real-world datasets. This section is devoted to explore, in the context of real datasets forecasting, whether NoVaS-type methods can provide good long-term time-aggregated forecasting ability and how our new methods are compared to the existing Model-free method.\n\nFor performing an extensive analysis and subsequently acquiring a convincing conclusion, we use three types of data--stock, index and currency data--to do predictions. Moreover, as done in simulation studies, we apply this exercise on two different lengths of data. For building large datasets (2-years period data), we take new data which come from Jan.2018 to Dec.2019 and old data which come from around 20 years ago, separately. The dynamics of these econometric datasets have changed a lot in the past 20 years and thus we wanted to explore whether our methods are good enough for both old and new data. Subsequently, we challenge our methods using short (1-year period) real-life data. Finally, we also do forecasting using volatile data, i.e., data from Nov. 2019 to Oct. 2020. Note that economies across the world went through a recession due to the COVID-19 pandemic and then slowly recovered during this time-period, typically these sort of situations introduce systematic perturbation in the dynamics of econometric datasets. We wanted to see if our methods can sustain such perturbations or abrupt changes. \n\n\\subsection{Old and new 2-years data}\\label{ssec:realdatanormalperiod2years}\n\\noindent For mimicking the 2-years period data, we adopt several stock datasets with 500 data size to do forecasting. In summary, we still compare different methods' performance on 1-step, 5-steps and 30-steps ahead POOS time-aggregated predictions. Performing the similar procedure as which we did in \\cref{sec:simu}, all results are shown in \\cref{6t1}. We can clearly find NoVaS-type methods still outperform the GARCH-direct method. Additionally, although the GE-NoVaS method is indistinguishable with the GA-NoVaS method, our new method is more robust than the GE-NoVaS method, see the 30-steps ahead prediction of old two-years BAC and MSFT cases. We can also notice that the GA-NoVaS-without-$\\beta$ method is more robust than other two NoVaS methods. The $\\beta$-removing idea proposed by \\cite{wu2021boosting} is substantiated again. \n\nSince the main goal of this article is offering a new type of NoVaS method which has better performance than the GE-NoVaS method for dealing with short and volatile data, we provide more extensive data analysis to support our new methods in next sections. \n\\begin{table}[htbp]\n \\caption{Comparison results of using old and new 2-years data}\n \\label{6t1}\n\\begin{adjustbox}{width=1\\textwidth}\n\\centering\n\\small\n\\begin{tabular}{lcccclcccc}\n \\toprule\n \\thead{Old \\\\2-years} & \\thead{\\small GE} & \\thead{\\small GA} & \\thead{\\small P-GA} & \\thead{\\small GARCH} & \\thead{New \\\\2-years}& \\thead{\\small GE} & \\thead{\\small GA} & \\thead{\\small P-GA} & \\thead{\\small GARCH} \\\\ \n \\midrule\n\n AAPL-1step & 0.99795 & 0.99236 & \\textbf{0.97836} & 1.00000 & AAPL-1step & 0.80150 & \\textbf{0.79899} & 0.79915 & 1.00000 \\\\[2pt]\n AAPL-5steps & 1.04919 & 1.04800 & \\textbf{0.96999} & 1.00000 & AAPL-5steps & 0.41405 & 0.42338 & \\textbf{0.40427} & 1.00000 \\\\[2pt]\n AAPL-30steps & 1.12563 & 1.21986 & \\textbf{0.96174} & 1.00000 & AAPL-30steps & \\textbf{0.13207} & 0.14046 & 0.14543 & 1.00000 \\\\[2pt]\n BAC-1step & \\textbf{0.99889} & 1.00396 & 1.02780 & 1.00000 & BAC-1step & 0.98393 & 0.99164 & \\textbf{0.96542} & 1.00000 \\\\[2pt]\n BAC-5steps & 1.04424 & 1.02185 & \\textbf{0.99399} & 1.00000 & BAC-5steps & 0.98885 & 1.01480 & \\textbf{0.91857} & 1.00000 \\\\[2pt]\n BAC-30steps & 1.32452 & 1.13887\\textbf{*} & 1.00363 & \\textbf{1.00000} & BAC-30steps & 1.14111 & 1.03657 & \\textbf{0.88596} & 1.00000 \\\\[2pt]\n MSFT-1step & 0.98785 & 0.98598 & \\textbf{0.96185} & 1.00000 & MSFT-1step & 0.98405 & 0.98630 & \\textbf{0.96374} & 1.00000 \\\\[2pt]\n MSFT-5steps & 1.00236 & 1.00096 & \\textbf{0.95271} & 1.00000 & MSFT-5steps & 0.65027 & 0.67005 & \\textbf{0.64278} & 1.00000 \\\\[2pt]\n MSFT-30steps & 1.25272 & 1.09881\\textbf{*} & \\textbf{0.88515} & 1.00000 & MSFT-30steps & \\textbf{0.19767} & 0.20060 & 0.21473 & 1.00000 \\\\[2pt]\n MCD-1step & 1.01845 & 1.00789 & \\textbf{0.99005} & 1.00000 & MCD-1step & 0.99631 & 0.99539 & \\textbf{0.98035} & 1.00000 \\\\[2pt]\n MCD-5steps & 1.11249 & 1.07748 & \\textbf{0.97777} & 1.00000 & MCD-5steps & 0.95403 & 0.95327 & \\textbf{0.91317} & 1.00000 \\\\[2pt]\n MCD-30steps & 1.76385 & 1.69757 & \\textbf{0.99418} & 1.00000 & MCD-30steps & 0.75730 & 0.75361 & \\textbf{0.74557} & 1.00000 \\\\[2pt]\n \\bottomrule\n \\end{tabular}\n \\end{adjustbox}\n \\\\\n \n \\tiny \\textit{Note:} Column names ``GA'' and ``GE'' represent GE-NoVaS and GA-NoVaS methods, respectively; ``GARCH'' means GARCH-direct method; ``P-GA'' means GA-NoVaS-without-$\\beta$ method. The benchmark is the GARCH-direct method, so numerical values in the table corresponding to GARCH-direct method are 1. Other numerical values are relative values compared to the GARCH-direct method. The bold value means that the corresponding method is the optimal choice for this data case. Cell with $*$ means the GA-NoVaS method is at least 10$\\%$ better than the GE-NoVaS method or inversely the GE-NoVaS method is at least 10$\\%$ better.\n\\end{table}\n\n\\subsection{2018 and 2019 1-year data }\\label{ssec:realdatanormalperiod1year}\n\\noindent For challenging our new methods in contrast to other methods for small real-life datasets, we separate every new 2-years period data in \\cref{ssec:realdatanormalperiod2years} to two 1-year period datasets, i.e., separate four new stock datasets to eight samples. We believe evaluating the prediction performance using shorter data is a more important problem and thus we wanted to make our analysis very comprehensive. Therefore, for this exercise, we add 7 index datasets: Nasdaq, NYSE, Small Cap, Dow Jones, S$\\&$P 500 , BSE and BIST; and two stock datasets: Tesla and Bitcoin into our analysis. \n\nFrom \\cref{6t2} which presents prediction results of different methods on 2018 and 2019 stock data, we still observe that NoVaS-type methods outperform GARCH-direct method for almost all cases. Among different NoVaS methods, it is clear that our new methods are superior than the existing GE-NoVaS method. For 30-steps ahead predictions of 2018-BAC data, 2019-MCD and Tesla data, etc, the existing NoVaS method is even worse than the GARCH-direct method. On the other hand, the GA-NoVaS method is more stable than the GE-NoVaS method, e.g., 30$\\%$ improvement is created for the 30-steps ahead prediction of 2018-BAC data. After applying the $\\beta$-removing idea, the GA-NoVaS-without-$\\beta$ significantly beats other methods for almost all cases.\n\nFrom \\cref{6t3} which presents prediction results of different methods on 2018 and 2019 index data, we can get the exactly same conclusion as before. NoVaS-type methods are far superior than the GARCH-direct and our new NoVaS methods outperform the existing GE-NoVaS method. Interestingly, the GE-NoVaS method is again beaten by the GARCH-direct method in some cases, such as 2019-Nasdaq, Smallcap and BIST. On the other hand, new methods still show more stable performance. Compared to the existing GE-NoVaS method, the GA-NoVaS-without-$\\beta$ method creates around 60$\\%$ improvement from the GE-NoVaS method on the 30-steps ahead prediction of 2019-BIST data. In addition, the GA-NoVaS method shows more than 10$\\%$ improvement for all 2018-BSE cases.\n\nCombining results presented in \\cref{6t1,6t2,6t3}, our new methods present better performance than existing GE-NoVaS and GARCH-direct methods on dealing with small and large real-life data. The improvement generated by new methods using shorter sample size (1-year data) is more significant than using larger sample size (2-years data).\n\n\n\\begin{table}[H]\n \\caption{Comparison results of using 2018 and 2019 stock data}\n \\label{6t2}\n \\begin{adjustbox}{width=1\\textwidth}\n\\small\n\\begin{tabular}{lcccclcccc}\n \\toprule \n 2018& \\thead{GE} & \\thead{GA} & \\thead{P-GA} & \\thead{GARCH} & 2019 & \\thead{GE} & \\thead{GA} & \\thead{P-GA} &\\thead{GARCH} \\\\\n \\midrule\n\n MCD-1step & 0.98514 & 0.97887 & \\textbf{0.94412} & 1.00000 & MCD-1step & 0.95959 & 0.96348 & \\textbf{0.94559} & 1.00000 \\\\[2pt]\n MCD-5steps & 1.0272 & 1.02519 & \\textbf{0.88151} & 1.00000 & MCD-5steps & 1.00723 & 1.01169 & \\textbf{0.90602} & 1.00000 \\\\[2pt]\n MCD-30steps & 0.62614 & 0.63992 & \\textbf{0.61153} & 1.00000 & MCD-30steps & 1.05239 & 0.95714 & \\textbf{0.77976} & 1.00000 \\\\[2pt]\n AAPL-1step & 0.92014 & 0.92317 & \\textbf{0.89283} & 1.00000 & AAPL-1step & 0.84533 & \\textbf{0.81326} & 0.81872 & 1.00000 \\\\[2pt]\n AAPL-5steps & 0.84798 & 0.73461\\textbf{*} & \\textbf{0.71233} & 1.00000 & AAPL-5steps & 0.85401 & 0.79254 & \\textbf{0.68792} & 1.00000 \\\\[2pt]\n AAPL-30steps & 0.38612 & \\textbf{0.36324} & 0.37081 & 1.00000 & AAPL-30steps & 0.99043 & 0.99286 & \\textbf{0.72892} & 1.00000 \\\\[2pt]\n BAC-1step & 0.94952 & 0.93842 & 0\\textbf{.92619} & 1.00000 & BAC-1step & 1.04272 & 1.04722 & \\textbf{0.98605} & 1.00000 \\\\[2pt]\n BAC-5steps & 0.83395 & 0.79158 & \\textbf{0.72512} & 1.00000 & BAC-5steps & 1.22761 & 1.20195 & \\textbf{0.95436} & 1.00000 \\\\[2pt]\n BAC-30steps & 1.34367 & 0.90675\\textbf{*} & \\textbf{0.8763} & 1.00000 & BAC-30steps & 1.4502 & 1.41788 & 1.03482 & \\textbf{1.00000} \\\\[2pt]\n MSFT-1step & 0.91705 & \\textbf{0.90936} & 0.95921 & 1.00000 & MSFT-1step & 1.03308 & 1.00101 & \\textbf{0.95347} & 1.00000 \\\\[2pt]\n MSFT-5steps & 0.74553 & 0.74267 & \\textbf{0.74237} & 1.00000 & MSFT-5steps & 1.2234 & 1.18205 & \\textbf{0.95417} & 1.00000 \\\\[2pt]\n MSFT-30steps & 0.6699 & 0.6477 & \\textbf{0.64717} & 1.00000 & MSFT-30steps & 1.2302 & 1.21337 & \\textbf{0.98476} & 1.00000 \\\\[2pt]\n Tesla-1step & 1.00181 & 0.96074 & \\textbf{0.86238} & 1.00000 & Tesla-1step & 1.00428 & 1.01934 & \\textbf{0.98955} & 1.00000 \\\\[2pt]\n Tesla-5steps & 1.20383 & 1.13335 & 1.0156 & \\textbf{1.00000} & Tesla-5steps & 1.0661 & 1.07506 & \\textbf{0.96107} & 1.00000 \\\\[2pt]\n Tesla-30steps & 1.97328 & 1.84871 & 1.25005 & \\textbf{1.00000} & Tesla-30steps & 2.00623 & 1.71782\\textbf{*} & \\textbf{0.84366} & 1.00000 \\\\[2pt]\n Bitcoin-1step & 0.99636 & 1.01731 & \\textbf{0.97734} & 1.00000 & Bitcoin-1step & 0.89929 & 0.88914 & \\textbf{0.87256} & 1.00000 \\\\[2pt]\n Bitcoin-5steps & 1.02021 & 1.1188 & \\textbf{0.93826} & 1.00000 & Bitcoin-5steps & 0.62312 & 0.63075 & \\textbf{0.56789} & 1.00000 \\\\[2pt]\n Bitcoin-30steps & \\textbf{0.86649} & 0.95506 & 0.91364 & 1.00000 & Bitcoin-30steps & 0.00733 & 0.00749 & \\textbf{0.00631} & 1.00000 \\\\[2pt]\n\n \\bottomrule\n \\end{tabular}\n \\end{adjustbox}\n \\\\\n \\tiny \\textit{Note:} Column names ``GA'' and ``GE'' represent GE-NoVaS and GA-NoVaS methods, respectively; ``GARCH'' means GARCH-direct method; ``P-GA'' means GA-NoVaS-without-$\\beta$ method. The benchmark is the GARCH-direct method, so numerical values in the table corresponding to GARCH-direct method are 1. Other numerical values are relative values compared to the GARCH-direct method. The bold value means that the corresponding method is the optimal choice for this data case. Cell with $*$ means the GA-NoVaS method is at least 10$\\%$ better than the GE-NoVaS method or inversely the GE-NoVaS method is at least 10$\\%$ better.\n\\end{table}\n\n\\begin{table}[H]\n \\caption{Comparison results of using 2018 and 2019 index data}\n \\setlength{\\abovecaptionskip}{0pt}\n \\label{6t3}\n \\begin{adjustbox}{width=1\\textwidth}\n\\small\n\\begin{tabular}{lcccclcccc}\n \\toprule \n 2018 & \\thead{GE} & \\thead{GA} & \\thead{P-GA} & \\thead{GARCH} & 2019& \\thead{GE} & \\thead{GA} & \\thead{P-GA} & \\thead{GARCH} \\\\ \n \\midrule\n Nasdaq-1step & \\textbf{0.91309} & 0.92303 & 0.92421 & 1.00000 & Nasdaq-1step & 0.99960 & 0.98950 & \\textbf{0.93843} & 1.00000 \\\\[2pt]\n Nasdaq-5steps & \\textbf{0.76419} & 0.79718 & 0.78823 & 1.00000 & Nasdaq-5steps & 1.15282 & 1.09176 & \\textbf{0.84051} & 1.00000 \\\\[2pt]\n Nasdaq-30steps & 0.66520 & \\textbf{0.65489} & 0.67389 & 1.00000 & Nasdaq-30steps & 0.68994 & 0.69846 & \\textbf{0.59218} & 1.00000 \\\\[2pt]\n NYSE-1step & 0.93509 & \\textbf{0.93401} & 0.96619 & 1.00000 & NYSE-1step & 0.92486 & \\textbf{0.91118} & 0.92193 & 1.00000 \\\\[2pt]\n NYSE-5steps & 0.83725 & 0.79330 & \\textbf{0.75822} & 1.00000 & NYSE-5steps & 0.86249 & 0.82114 & \\textbf{0.71038} & 1.00000 \\\\[2pt]\n NYSE-30steps & 0.75053 & \\textbf{0.61443*} & 0.61830 & 1.00000 & NYSE-30steps & 0.22122 & 0.22173 & \\textbf{0.18116} & 1.00000 \\\\[2pt]\n Smallcap-1step & \\textbf{0.90546} & 0.91346 & 0.91101 & 1.00000 & Smallcap-1step & 1.02041 & 1.00626 & \\textbf{0.98482} & 1.00000 \\\\[2pt]\n Smallcap-5steps & \\textbf{0.72627} & 0.73955 & 0.73223 & 1.00000 & Smallcap-5steps & 1.15868 & 1.08929 & \\textbf{0.85490} & 1.00000 \\\\[2pt]\n Samllcap-30steps & 0.50005 & 0.46482 & \\textbf{0.46312} & 1.00000 & Samllcap-30steps & 1.30467 & 1.28949 & \\textbf{0.90360} & 1.00000 \\\\[2pt]\n Djones-1step & 0.90932 & \\textbf{0.90707} & 0.91192 & 1.00000 & Djones-1step & 0.96752 & \\textbf{0.96433} & 0.96977 & 1.00000 \\\\[2pt]\n Djones-5steps & 0.82480 & 0.79965 & \\textbf{0.76226} & 1.00000 & Djones-5steps & 0.98725 & 0.93315 & \\textbf{0.91238} & 1.00000 \\\\[2pt]\n Djones-30steps & 0.72547 & \\textbf{0.53021*} & 0.56854 & 1.00000 & Djones-30steps & 0.86333 & 0.85006 & \\textbf{0.81803} & 1.00000 \\\\[2pt]\n SP500-1step & 0.91860 & 0.91256 & \\textbf{0.88405} & 1.00000 & SP500-1step & 0.96978 & 0.96526 & \\textbf{0.93162} & 1.00000 \\\\[2pt]\n SP500-5steps & 0.85108 & 0.77305 & \\textbf{0.75646} & 1.00000 & SP500-5steps & 0.96704 & 0.94028 & \\textbf{0.77434} & 1.00000 \\\\[2pt]\n SP500-30steps & 0.88917 & \\textbf{0.68156*} & 0.72104 & 1.00000 & SP500-30steps & 0.34389 & 0.34537 & \\textbf{0.30127} & 1.00000 \\\\[2pt]\n BSE-1step & 0.99942 & \\textbf{0.88322*} & 0.92568 & 1.00000 & BSE-1step & 0.70667 & 0.70194 & \\textbf{0.66667} & 1.00000 \\\\[2pt]\n BSE-5steps & 0.92061 & \\textbf{0.78484*} & 0.84408 & 1.00000 & BSE-5steps & 0.25675 & 0.25897 & \\textbf{0.23603} & 1.00000 \\\\[2pt]\n BSE-30steps & 0.52431 & \\textbf{0.41010*} & 0.44092 & 1.00000 & BSE-30steps & 0.03764 & 0.03951 & \\textbf{0.02888} & 1.00000 \\\\[2pt]\n BIST-1step & 0.93221 & \\textbf{0.92215} & 0.94138 & 1.00000 & BIST-1step & \\textbf{0.96807} & 0.97209 & 0.98234 & 1.00000 \\\\[2pt]\n BIST-5steps & 0.82149 & \\textbf{0.79664} & 0.81417 & 1.00000 & BIST-5steps & 0.98944 & 1.03903 & \\textbf{0.85370} & 1.00000 \\\\[2pt]\n BIST-30steps & 1.34581 & 1.42233 & 1.09900 & \\textbf{1.00000} & BIST-30steps & 2.21996 & 2.10562 & \\textbf{0.85743} & 1.00000 \\\\[2pt]\n\n \\bottomrule\n \\end{tabular}\n \\end{adjustbox}\n \\\\\n \\tiny \\textit{Note:} Column names ``GA'' and ``GE'' represent GE-NoVaS and GA-NoVaS methods, respectively; Column name ``GARCH'' means GARCH-direct method; ``P-GA'' means GA-NoVaS-without-$\\beta$ method. The benchmark is the GARCH-direct method, so numerical values in the table corresponding to GARCH-direct method are 1. Other numerical values are relative values compared to the GARCH-direct method. The bold value means that the corresponding method is the optimal choice for this data case. Cell with $*$ means the GA-NoVaS method is at least 10$\\%$ better than the GE-NoVaS method or inversely the GE-NoVaS method is at least 10$\\%$ better.\n\\end{table}\n\n\\subsection{Volatile 1-year data}\\label{ssec: realdatavolatileperiod}\n\\noindent In this subsection, we perform POOS forecasting using volatile 1-year data (i.e., data from Nov. 2019 to Oct. 2020). We tactically choose this period data to challenge our new methods for checking whether it can self-adapt to the structural incoherence between pre- and post-pandemic, and we also want to compare our new methods with the existing GE-NoVaS method. For observing affects of pandemic, we can take the price of SP500 index as an example. From \\cref{6f1}, it is clearly that the price grew slowly during the normal period form Jan. 2017 to Dec. 2017. However, during the most recent one year, the price fluctuated severely due to the pandemic. \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=9cm,height=6cm]{contrastofsp500.eps}\n\\caption{The left subfigure depicts the price of SP500 from Jan.2017 to Dec.2017 which presents a slow growth; The right subfigure depicts the price of SP500 from Nov.2019 to Oct.2020}\n\\label{6f1}\n\\end{figure}\n\nSimilarly, we focus on evaluating the performance of NoVaS-type methods on handling volatile data by doing comparisons with the GARCH-direct method. For executing a comprehensive analysis, we again investigate different methods' performance on stock, index and currency data. \n\n\\subsubsection{Stock data}\\label{sssec:stock}\n\\noindent The POOS forecasting results of volatile 1-year stock datasets are presented in \\cref{6t4}. NoVaS-type methods dominate the GARCH-direct method. The performance of the GARCH-direct method is terrible especially for the Bitcoin case. Apart from this overall advantage of NoVaS-type methods, there is no doubt that the GA-NoVaS method manifests greater prediction results than the GE-NoVaS method since it occupies 13 out 27 optimal choices and stands at least 10$\\%$ improvement for 5 cases. The parsimonious GA-NoVaS-without-$\\beta$ also shows better results than the GE-NoVaS method. This phenomenon lends strong evidence to support our postulation that the GA-NoVaS method is more appropriate to handle volatile data. \n\n\\begin{table}[htbp]\n \\caption{Comparison results of using volatile 1-year stock data}\n \\label{6t4}\n \n \\centering\n\\scriptsize\n\\begin{tabular}{lcccccccc}\n \\toprule \n & \\thead{\\scriptsize GE-NoVaS} & \\thead{\\scriptsize GA-NoVaS} & \\thead{\\scriptsize GA-NoVaS-without-$\\beta$} & \\thead{\\scriptsize GARCH-direct} \\\\ \n \\midrule\n NKE-1step & 0.63568 & \\textbf{0.63209} & 0.65594 & 1.00000 \\\\ [1.2pt]\n NKE-5steps & 0.20171 & \\textbf{0.19089} & 0.22226 & 1.00000 \\\\ [1.2pt]\n NKE-30steps & 0.00411 & \\textbf{0.00278*} & 0.00340 & 1.00000\\\\[1.2pt]\n AMZN-1step & 0.97099 & 0.96719 & \\textbf{0.90487} & 1.00000 \\\\[1.2pt]\n AMZN-5steps & 0.88705 & 0.88274 & \\textbf{0.72850} & 1.00000\\\\[1.2pt]\n AMZN-30steps & 0.58124 & 0.62863 & \\textbf{0.53310} & 1.00000\\\\[1.2pt]\n IBM-1step & 0.80222 & 0.79823 & \\textbf{0.79509} & 1.00000 \\\\[1.2pt]\n IBM-5steps & 0.38933 & \\textbf{0.37346} & 0.38413 & 1.00000 \\\\[1.2pt]\n IBM-30steps & 0.01143 & 0.00996\\textbf{*} & \\textbf{0.00879} & 1.00000 \\\\[1.2pt]\n MSFT-1step & 0.80133 & \\textbf{0.79528} & 0.81582 & 1.00000 \\\\[1.2pt]\n MSFT-5steps & 0.35567 & \\textbf{0.33419} & 0.38022 & 1.00000 \\\\[1.2pt]\n MSFT-30steps & 0.01342 & 0.01031\\textbf{*} & \\textbf{0.00784} & 1.00000 \\\\[1.2pt]\n SBUX-1step & 0.68206 & 0.67067 & \\textbf{0.66743} & 1.00000 \\\\[1.2pt]\n SBUX-5steps & 0.24255 & \\textbf{0.23072} & 0.26856 & 1.00000 \\\\[1.2pt]\n SBUX-30steps & 0.00499 & 0.00337\\textbf{*} & \\textbf{0.00236} & 1.00000 \\\\[1.2pt]\n KO-1step & 0.77906 & \\textbf{0.75389} & 0.77035 & 1.00000 \\\\[1.2pt]\n KO-5steps & 0.34941 & \\textbf{0.32459} & 0.33405 & 1.00000 \\\\[1.2pt]\n KO-30steps & 0.01820 & 0.01848 & \\textbf{0.01582} & 1.00000 \\\\[1.2pt]\n MCD-1step & 0.51755 & \\textbf{0.51351} & 0.56414 & 1.00000 \\\\[1.2pt]\n MCD-5steps & 0.10725 & \\textbf{0.09714} & 0.17439 & 1.00000 \\\\[1.2pt]\n MCD-30steps & 3.32E-05 & 2.97E-05\\textbf{*} & \\textbf{7.62E-06} & 1.00000 \\\\[1.2pt]\n Tesla-1step & 0.90712 & 0.90250 & \\textbf{0.88782} & 1.00000 \\\\[1.2pt]\n Tesla-5steps & 0.68450 & 0.67935 & \\textbf{0.66937} & 1.00000 \\\\[1.2pt]\n Tesla-30steps & \\textbf{0.21643} & 0.21718 & 0.22395 & 1.00000 \\\\[1.2pt]\n Bitcoin-1step & 0.36323 & \\textbf{0.36260} & 0.36326 & 1.00000 \\\\[1.2pt]\n Bitcoin-5steps & \\textbf{0.01319} & 0.01321 & 0.01322 & 1.00000 \\\\[1.2pt]\n Bitcoin-30steps & 7.75E-17 & \\textbf{7.65E-17} & 7.75E-17 & 1.00000 \\\\[1.2pt]\n \\bottomrule\n \\end{tabular}\n \n \\\\\n \\raggedright\n \\tiny \\textit{Note:} The benchmark is the GARCH-direct method, so numerical values in the table corresponding to GARCH-direct method are 1. Other numerical values are relative values compared to the GARCH-direct method. The bold value means that the corresponding method is the optimal choice for this data case. Cell with $*$ means the GA-NoVaS method is at least 10$\\%$ better than the GE-NoVaS method or inversely the GE-NoVaS method is at least 10$\\%$ better.\n\\end{table}%\n\n\\subsubsection{Currency data}\\label{sssec:currency}\n\\noindent The POOS forecasting results of most recent 1-year currency datasets are presented in \\cref{6t5}. One thing should be noticed is that \\citet{fryzlewicz2008normalized} implied the ARCH framework seems to be a superior methodology for dealing with the currency exchange data. Therefore, we should not anticipate that GA-NoVaS-type methods can attain much improvement for this data case. However, the GA-NoVaS method still brings off around 26$\\%$ and 37$\\%$ improvement for 30-steps ahead predictions of CADJPY and CNYJPY, respectively. Besides, the GA-NoVaS-without-$\\beta$ method also remains great performance. This surprising result can be seen as an evidence to show GA-NoVaS-type methods are robust to model misspecification.\n\n\\begin{table}[htbp]\n \\caption{Comparison results of using volatile 1-year currency data}\n \\label{6t5}\n \\centering\n \n\\scriptsize\n\\begin{tabular}{lcccccccc}\n \\toprule \n & \\thead{\\scriptsize GE-NoVaS} & \\thead{\\scriptsize GA-NoVaS} & \\thead{\\scriptsize GA-NoVaS-without-$\\beta$} & \\thead{\\scriptsize GARCH-direct} \\\\ \n \\midrule\n CADJPY-1step & 0.46940 & \\textbf{0.46382} & 0.48367 & 1.00000 \\\\[1.2pt]\n CADJPY-5steps & 0.11678 & \\textbf{0.11620} & 0.14376 & 1.00000 \\\\[1.2pt]\n CADJPY-30steps & 0.00584 & \\textbf{0.00430*} & 0.00482 & 1.00000 \\\\[1.2pt]\n EURJPY-1step & 0.95093 & \\textbf{0.94682} & 0.95133 & 1.00000 \\\\[1.2pt]\n EURJPY-5steps & 0.76182 & 0.77091 & \\textbf{0.75636} & 1.00000 \\\\[1.2pt]\n EURJPY-30steps & \\textbf{0.16202} & 0.17956 & 0.18189 & 1.00000 \\\\[1.2pt]\n USDCNY-1step & 0.98905 & 0.97861 & \\textbf{0.95757} & 1.00000 \\\\[1.2pt]\n USDCNY-5steps & 0.93182 & 0.92614 & \\textbf{0.83523} & 1.00000 \\\\[1.2pt]\n USDCNY-30steps & 0.57171 & \\textbf{0.57100} & 0.60131 & 1.00000 \\\\[1.2pt]\n GBPJPY-1step & 0.86971 & \\textbf{0.86474} & 0.87160 & 1.00000 \\\\[1.2pt]\n GBPJPY-5steps & 0.49749 & 0.49612 & \\textbf{0.48842} & 1.00000 \\\\[1.2pt]\n GBPJPY-30steps & 0.17058 & \\textbf{0.16987} & 0.17262 & 1.00000 \\\\[1.2pt]\n USDINR-1step & 0.97289 & 0.96829 & \\textbf{0.93140} & 1.00000 \\\\[1.2pt]\n USDINR-5steps & 0.80866 & 0.78008 & \\textbf{0.75693} & 1.00000 \\\\[1.2pt]\n USDINR-30steps & \\textbf{0.09725} & 0.09889 & 0.11380 & 1.00000 \\\\[1.2pt]\n CNYJPY-1step & 0.77812 & 0.77983 & \\textbf{0.74586} & 1.00000 \\\\[1.2pt]\n CNYJPY-5steps & 0.38875 & 0.38407 & \\textbf{0.34839} & 1.00000 \\\\[1.2pt]\n CNYJPY-30steps & 0.08398 & \\textbf{0.05240*} & 0.05444 & 1.00000 \\\\[1.2pt]\n \\bottomrule\n \\end{tabular}\n \n\\end{table}%\n\n\n\\subsubsection{Index data}\\label{sssec:index}\n\\noindent The POOS forecasting results of most recent 1-year index datasets are presented in \\cref{6t6}. Consistent with conclusions corresponding to previous two classes of data, NoVaS-type methods still have obviously better performance than the GARCH-direct method. Besides this advantage of NoVaS methods, new methods still govern the existing GE-NoVaS method. In addition to these expected results, we find the GE-NoVaS method is even 14$\\%$ worse than the GARCH-direct method for 1-step USDX future case. On the other hand, GA-NoVaS-type methods still keep great performance. This phenomenon also appears in \\cref{sssec:simuresultsmoeld1-5,sssc:simudifferent,ssc:simusmall,ssec:realdatanormalperiod2years,ssec:realdatanormalperiod1year}. Beyond this, there are 12 cases that the GA-NoVaS method renders more than 10$\\%$ improvement compared to the GE-NoVaS method. A most significant case is the 30-steps ahead prediction of Bovespa data where around 60$\\%$ improvement is introduced by the GA-NoVaS method compared with GE-NoVaS method. \n\\begin{table}[htbp]\n \\caption{Comparison results of using volatile 1-year index data}\n \\label{6t6}\n \\centering\n \n\\scriptsize\n\\begin{tabular}{lcccccccc}\n \\toprule \n & \\thead{\\scriptsize GE-NoVaS} & \\thead{\\scriptsize GA-NoVaS} & \\thead{\\scriptsize GA-NoVaS-without-$\\beta$} & \\thead{\\scriptsize GARCH-direct} \\\\ \n \\midrule\n SP500-1step & 0.97294 & 0.95881 & \\textbf{0.92854} & 1.00000 \\\\[1.2pt]\n SP500-5steps & 0.96590 & 0.94457 & \\textbf{0.77060} & 1.00000 \\\\[1.2pt]\n SP500-30steps & 0.34357 & 0.34561 & \\textbf{0.30115} & 1.00000 \\\\[1.2pt]\n Nasdaq-1step & 0.71380 & \\textbf{0.70589} & 0.77753 & 1.00000 \\\\[1.2pt]\n Nasdaq-5steps & 0.29332 & \\textbf{0.27007} & 0.36428 & 1.00000 \\\\[1.2pt]\n Nasdaq-30steps & 0.01223 & \\textbf{0.00618*} & 0.00696 & 1.00000 \\\\[1.2pt]\n NYSE-1step & 0.55741 & 0.55548 & \\textbf{0.54598} & 1.00000 \\\\[1.2pt]\n NYSE-5steps & 0.08994 & \\textbf{0.07666*} & 0.07798 & 1.00000 \\\\[1.2pt]\n NYSE-30steps & 1.36E-05 & 9.06E-06\\textbf{*} & \\textbf{6.57E-06} & 1.00000 \\\\[1.2pt]\n Smallcap-1step & 0.58170 & \\textbf{0.57392} & 0.57773 & 1.00000 \\\\[1.2pt]\n Smallcap-5steps & 0.10270 & 0.10135 & \\textbf{0.09628} & 1.00000 \\\\[1.2pt]\n Smallcap-30steps & 7.00E-05 & 4.33E-05\\textbf{*} & \\textbf{3.65E-05} & 1.00000 \\\\[1.2pt]\n BSE-1step & 0.39493 & \\textbf{0.37991} & 0.39851 & 1.00000 \\\\[1.2pt]\n BSE-5steps & 0.03320 & \\textbf{0.02829*} & 0.04170 & 1.00000 \\\\[1.2pt]\n BSE-30steps & 2.45E-05 & 2.19E-05\\textbf{*} & \\textbf{1.73E-05} & 1.00000 \\\\[1.2pt]\n DAX-1step & \\textbf{0.65372} & 0.65663 & 0.66097 & 1.00000 \\\\[1.2pt]\n DAX-5steps & 0.10997 & \\textbf{0.10828} & 0.11085 & 1.00000 \\\\[1.2pt]\n DAX-30steps & 4.97E-05 & \\textbf{4.87E-05} & 7.81E-05 & 1.00000\\\\[1.2pt]\n USDX future-1step & 1.14621 & 1.00926\\textbf{*} & 1.03693 & \\textbf{1.00000} \\\\[1.2pt]\n USDX future-5steps & 0.61075 & 0.53834\\textbf{*} & \\textbf{0.51997} & 1.00000 \\\\[1.2pt]\n USDX future-30steps & 0.10723 & \\textbf{0.09911} & 0.10063 & 1.00000 \\\\[1.2pt]\n Bovespa-1step & 0.60031 & \\textbf{0.57316} & 0.60656 & 1.00000 \\\\[1.2pt]\n Bovespa-5steps & 0.08603 & \\textbf{0.06201*} & 0.09395 & 1.00000 \\\\[1.2pt]\n Bovespa-30steps & 6.87E-06 & \\textbf{2.82E-06*} & 3.19E-06 & 1.00000 \\\\[1.2pt]\n Djones-1step & 0.56357 & 0.55020 & \\textbf{0.54422} & 1.00000 \\\\[1.2pt]\n Djones-5steps & 0.09810 & \\textbf{0.08239*} & 0.08698 & 1.00000 \\\\[1.2pt]\n Djones-30steps & 4.32E-05 & \\textbf{2.22E-05*} & 2.65E-05 & 1.00000 \\\\[1.2pt]\n BIST-1step & 0.94794 & 0.95313 & \\textbf{0.92418} & 1.00000 \\\\[1.2pt]\n BIST-5steps & \\textbf{0.48460} & 0.49098 & 0.49279 & 1.00000 \\\\[1.2pt]\n BIST-30steps & \\textbf{0.05478} & 0.05980 & 0.05671 & 1.00000 \\\\[1.2pt]\n \\bottomrule\n \\end{tabular}\n \n\\end{table}\n\n\\subsection{Summary of real-world data analysis}\\label{ssec:summaryofrealdataanalysis}\n\\noindent After performing extensive real-world data analysis, we can conclude that NoVaS-type methods have generally better performance than the GARCH-direct method. Sometimes, the long-term prediction of GARCH-direct method is impaired due to accumulated errors. Applying NoVaS-type methods can avoid such issue. In addition to this encouraging result, two new NoVaS methods proposed in this article all have greater performance than the existing GE-NoVaS method, especially for analyzing short and volatile data. The satisfactory performance of NoVaS-type methods on predicting Bitcoin data may also open up the application of using NoVaS-type methods to forecast cryptocurrency data. \n\n\n\\section{Comparison of predictive accuracy}\\label{sec:comparisonofpredictive}\nAs illustrated in \\cref{sec:intro}, accurate and robust volatility forecasting is an important focus for econometricians. Typically, volatility of returns can be characterized by GARCH-type models. Then, with the Model-free Prediction Principle being proposed, a more accurate NoVaS method was built to predict volatility. This paper further improves the existing NoVaS method by proposing a new transformation structure in \\cref{sec:method}. After performing extensive POOS predictions on different classes of data, we find our new methods achieve better prediction performance than traditional GARCH(1,1) model and the existing GE-NoVaS method. The most successful method is the GA-NoVaS-without-$\\beta$ method. \n\nHowever, one may still think the victory of our new methods is just caused by using specific sample even new methods show lower prediction error (i.e., calculated by \\cref{eq:4.1}) for almost all cases. Therefore, we want to learn whether this victory is statistically\nsignificant. We shall notice that \\cite{wu2021boosting} applied CW-tests to show removing-$\\beta$ idea is appropriate to refine the GE-NoVaS method. Likewise, we are curious about if this refinement is again reasonable for deriving the GA-NoVaS-without-$\\beta$ method from the GA-NoVaS method. In this paper, we focus on the CW-test built by \\cite{clark2007approximately}\\footnote{See \\cite{clark2007approximately} for theoretical details of this test, explaining these details is not in the scope of this paper.} which applied an adjusted Mean Squared Prediction Error (MSPE) statistics to test if parsimonious null model and larger model have equal predictive accuracy, see \\cite{dangl2012predictive,kong2011predicting,dai2021predicting} for examples of applying this CW-test.\n\n\n\n\\subsection{CW-test}\nNote that the GA-NoVaS-without-$\\beta$ method is a parsimonious method compared with the GA-NoVaS method. The reason of removing the $\\beta$ term has been illustrated in \\cref{ssecmotivation}. Here, we want to deploy the CW-test to make sure the $\\beta$-removing idea is not only empirically adoptable but also statistically reasonable. We take several results from \\cref{sec:real data} to run CW-tests. However, it is tricky to apply the CW-test on comparing 5-steps and 30-steps aggregated predictions. In other words, the CW-test result for aggregated predictions is ambiguous. It is hard to explain the meaning of a significant small $p$-value. Does this mean a method outperforms the opposite one for all single-step horizons? Or does this mean the method just achieves better performance at some specific future steps? Therefore, we just consider 1-step ahead prediction horizon and CW-test results are tabulated in \\cref{7t1}. \n\nFrom \\cref{7t1}, under a one-sided 5$\\%$ significance level, there is only 1 case out of total 28 cases which rejects the null hypothesis. Besides, we should notice that the CW-test still accepts the null hypothesis for 2018-MSFT and volatile period of MCD even the GA-NoVaS method has a better performance value on these cases. Moreover, the GA-NoVaS-without-$\\beta$ is more computationally efficient than the GA-NoVaS method. In summary, the reasonability of removing $\\beta$ term is shown again by comparing GA-NoVaS and GA-NoVaS-without-$\\beta$ methods. \n\n\\begin{table}[H]\n\\centering\n \\caption{CW-tests on 1-step ahead prediction of GA-NoVaS and GA-NoVaS-without-$\\beta$ methods}\n \\label{7t1}\n \\scriptsize \n\\begin{tabular}{lccc}\n \\toprule \n & \\thead{\\scriptsize P-value} & \\thead{\\scriptsize GA-NoVaS\\\\ \\scriptsize Performance} & \\thead{\\scriptsize GA-NoVaS-without-$\\beta$ \\\\ \\scriptsize Performance} \\\\\n \\midrule\n 2018-AAPL-1step & 0.99 & 0.92 & 0.89 \\\\ \n 2019-AAPL-1step & 0.08 & 0.81 & 0.82 \\\\ \n 2018-BAC-1step & 0.63 & 0.94 & 0.93 \\\\ \n 2019-BAC-1step & 0.49 & 1.05 & 0.99 \\\\ \n 2018-TSLA-1step & 0.27 & 0.92 & 0.86 \\\\ \n 2019-TSLA-1step & 0.22 & 1.02 & 0.99 \\\\\n 2018-MCD-1step & 0.57 & 0.98 & 0.94 \\\\ \n 2019-MCD-1step & 0.19 & 0.96 & 0.95 \\\\\n 2018-MSFT-1step & 0.17 & 0.91 & 0.96 \\\\\n 2019-MSFT-1step & 0.47 & 1.00 & 0.95 \\\\ \n 2018-Djones-1step & 0.64 & 0.91 & 0.91 \\\\ \n 2019-Djones-1step & 0.27 & 0.96 & 0.97 \\\\ \n 2018-Nasdaq-1step & 0.51 & 0.92 & 0.92 \\\\ \n 2019-Nasdaq-1step & 0.48 & 0.99 & 0.94 \\\\ \n 2018-NYSE-1step & 0.31 & 0.93 & 0.97 \\\\ \n 2019-NYSE-1step & 0.11 & 0.91 & 0.92 \\\\ \n 2018-SP500-1step & 0.42 & 0.91 & 0.88 \\\\ \n 2019-SP500-1step & 0.32 & 0.97 & 0.93 \\\\ \n 11.2019$\\sim$10.2020-IBM-1step & 0.26 & 0.80 & 0.80 \\\\ \n 11.2019$\\sim$10.2020-KO-1step & 0.01 & 0.75 & 0.77 \\\\ \n 11.2019$\\sim$10.2020-MCD-1step & 0.14 & 0.51 & 0.56 \\\\ \n 11.2019$\\sim$10.2020-SBUX-1step & 0.18 & 0.67 & 0.67 \\\\ \n 11.2019$\\sim$10.2020-CADJPY-1step & 0.07 & 0.46 & 0.48 \\\\ \n 11.2019$\\sim$10.2020-CNYJPY-1step & 0.66 & 0.78 & 0.75 \\\\ \n 11.2019$\\sim$10.2020-USDCNY-1step & 0.36 & 0.98 & 0.96 \\\\ \n 11.2019$\\sim$10.2020-EURJP-1step & 0.19 & 0.95 & 0.95 \\\\ \n 11.2019$\\sim$10.2020-Djones-1step & 0.30 & 0.56 & 0.55 \\\\ \n 11.2019$\\sim$10.2020-SP500-1step & 0.25 & 0.59 & 0.58 \\\\ \n\n \\bottomrule\n \\end{tabular}\\\\\n \\tiny\n \\raggedright\n \\textit{Note:} The null hypothesis of the CW-test is that parsimonious and larger models have equal MSPE. The alternative is that the larger model has a smaller MSPE. The performance of GA-NoVaS and GA-NoVaS-without-$\\beta$ methods are calculated as we did in \\cref{sec:real data}, which are relative values compared with benchmark method (GARCH-direct method).\n\\end{table}\n\n\\section{Conclusion}\\label{sec:conclusion}\n\\noindent In this paper, we show the current state-of-the-art GE-NoVaS and our proposed new methods can avoid error accumulation problem even when long-step ahead predictions are required. These methods outperform GARCH(1,1) model on predicting either simulated data or real-world data under different forecasting horizons. Moreover, the newly proposed GA-NoVaS method is a more stable structure to handle volatile and short data than the GE-NoVaS method. It can also bring significant improvement when the long-term prediction is desired. Additionally, although we reveal that parsimonious variants of GA-NoVaS and GE-NoVaS indeed possess a same structure, the GA-NoVaS-without-$\\beta$ method is still more favorable since the corresponding region of model parameter is more complete by design. In summary, the approach to build the NoVaS transformation through the GARCH(1,1) model is sensible and results in superior GA-NoVaS-type methods.\n\nIn the future, we plan to explore the NoVaS method in different directions. Our new methods corroborate that and also open up avenues where one can explore other specific transformation structures. In the financial market, the stock data move together. So it would be exciting to see if one can do Model-free predictions in a multiple time series scenario. In some areas, integer-valued time series has important applications. Thus, adjusting such Model-free predictions to deal with count data is also desired. There are also a lot of scopes in proving statistical validity of such predictions. First, we hope a rigorous and systematic way to compare predictive accuracy of NoVaS-type and standard GARCH method can be built. From a statistical inference point of view, one can also construct prediction intervals for these predictions using bootstrap. Such prediction intervals are well sought in the econometrics literature and some results on asymptotic validity of these can be proved. We can also explore dividing the dataset into test and training in some optimal way and see if that can improve performance of these methods. Additionally, since determining the transformation function involves optimization of unknown coefficients, designing a more efficient and precise algorithm may be a further direction to improve NoVaS-type methods. \n\\section{Acknowledgement}\\label{sec:ackno}\nThe first author is thankful to Professor Politis for introduction to the topic and useful discussions. The second author's research is partially supported by NSF-DMS 2124222.\n\n\\section{Data Availability Statement}\\label{sec:dataav}\nWe have collected all data presented here from \\url{www.investing.com} manually. Then, we transform the closing price data to financial log-returns based on \\cref{Eq:4.1}.\n\n\n\\bibliographystyle{spbasic}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{intro}\n\nLet $(\\mathsf{M},\\mathsf{J})$ be a compact, connected, \\emph{almost complex} manifold of dimension $2n$ and let $\\mathsf{c}=\\sum_{i=1}^n \\mathsf{c}_i$ be the total Chern class of the tangent bundle of $\\mathsf{M}$. \nTo each partition of $n$ one can associate an integer, called \\emph{Chern number}, given by\n$\\mathsf{c}_{i_1}\\cdots \\mathsf{c}_{i_k}[\\mathsf{M}]:=\\langle \\mathsf{c}_{i_1}\\cdots \\mathsf{c}_{i_k}, \\mu_\\mathsf{M}\\rangle$, where $i_1+\\cdots +i_k=n$ and $\\mu_\\mathsf{M}$ is the orientation homology class of $\\mathsf{M}$ (the orientation being induced by the almost complex structure). \nThe problem of determining which lists of integers can arise \nas the Chern numbers of a compact almost complex manifold $(\\mathsf{M},\\mathsf{J})$ of a given dimension (also known as the \\emph{geography} problem) has been investigated in different settings. Without additional assumptions on $(\\mathsf{M},\\mathsf{J})$, a theorem of Milnor \\cite{H} implies that it is necessary and sufficient for these integers to satisfy a certain set of congruences depending on $n$ (the same is true if $\\mathsf{M}$ is connected and $n\\geq 2$, see \\cite{Ge}). \nHowever, if the manifold is endowed with a $\\mathsf{J}$-preserving circle action, further restrictions arise and the geography problem is, in its generality, still open. When the fixed point set is empty (or more generally when all the stabilisers are discrete), as a consequence of the Atiyah--Bott--Berline--Vergne localization formula (hereinafter the ABBV formula, see Thm.\\ \\ref{abbv formula}) all the Chern numbers must vanish. \nIn this note we are interested\nin the case in which the fixed point set is non-empty and discrete (for recent results concerning non-isolated fixed points\nsee \\cite{Ku}). \n\n\\emph{Henceforth, the triple $(\\mathsf{M},\\mathsf{J},S^1)$ will denote a compact, connected, almost complex manifold acted on by a circle $S^1$ that preserves $\\mathsf{J}$, with nonempty, discrete fixed point set $\\mathsf{M}^{S^1}$, and will be referred to as an} {\\bf $S^1$-space}. \n\nThe Chern numbers of $S^1$-spaces satisfy more restrictions. \nFor instance, as a consequence of the ABBV formula, it is easy to see that \n$\\mathsf{c}_n[\\mathsf{M}] = \\chi(\\mathsf{M}) = |\\mathsf{M}^{S^1}|$,\nthus implying that $\\mathsf{c}_n[\\mathsf{M}]>0$, which is not true in general for any almost complex manifold $(\\mathsf{M},\\mathsf{J})$.\nIn 1979 Kosniowski \\cite{Ko} conjectured that the number of fixed points, and hence $\\mathsf{c}_n[\\mathsf{M}]$, grows linearly with $n$; more precisely\nhe predicted that $\\mathsf{c}_n[\\mathsf{M}] \\geq \\left \\lceil{\\frac{n}{2}}\\right \\rceil$. Even if much progress has been done to prove Kosniowski's conjecture (see \\cite{Ha}, and more recently \\cite{LL,LT,PT,CKP,GPS,J}), a complete answer is still missing. This\n shows that the geography problem for an $S^1$-space $(\\M,\\J,S^1)$ is much harder, and\nthe following questions naturally arise:\n\\begin{question}\\label{conj 1}\nWhat are all the possible values of the Chern numbers of $(\\M,\\J,S^1)$? Are there other (combinations of) Chern numbers satisfying (in)equalities depending on $n$?\n\\end{question}\nThe first goal of this note is to show that the Chern numbers of $(\\M,\\J,S^1)$ satisfy equations that depend on two integers,\nthe \\emph{index} $\\k0$ of $(\\mathsf{M},\\mathsf{J})$, and an integer $N_0$ defined by the action, see below. The second is to apply these results to symplectic manifolds supporting symplectic circle actions with discrete fixed point set, \nshowing `rigidity' results for the Chern numbers, and deriving topological conditions which ensure the manifold can only support Hamiltonian or only non-Hamiltonian actions, see Section \\ref{atsm}.\n\n Let $\\mathsf{c}_1\\in H^2(\\mathsf{M};{\\mathbb{Z}})$ be the first Chern class of the tangent bundle. The \\emph{index} $\\k0$ of $(\\mathsf{M},\\mathsf{J})$ is defined to be the largest integer such that, modulo torsion, $\\mathsf{c}_1=\\k0\\,\\eta_0$ for some non-zero element $\\eta_0\\in H^2(\\mathsf{M};{\\mathbb{Z}})$. In other words, $\\k0=0$ if $\\mathsf{c}_1$ is torsion, and is otherwise the biggest integer\n such that $\\mathsf{c}_1\/\\k0\\in H^2(\\mathsf{M};{\\mathbb{Z}})$, modulo torsion elements.\nWhen $\\mathsf{M}$ is simply connected and symplectic, the index coincides with the \\emph{minimal Chern number} (see Remark \\ref{mcn}).\nNote that $\\mathsf{M}$ is simply connected if it is endowed with a Hamiltonian circle action with isolated fixed points, see \\cite{Li2}.\nThe other integer $N_0$ depends on the action, and is defined as \nthe \\emph{number of fixed points with $0$ negative weights} (see Section \\ref{background} \\eqref{weights def}). \n\nWhen $\\k0=0$, namely when $\\mathsf{c}_1$ is a torsion element, all the Chern numbers involving the first Chern class, as well as the Todd genus (see Lemma \\ref{c1 N0} (a2)), must vanish.\n\n In this paper we are interested in\nanalysing what happens when $\\k0>0$, and a careful analysis is carried out when $\\k0\\geq n-2$. \nWhen $\\mathsf{c}_1$ is not torsion, the aforementioned equations among the Chern numbers of $(\\M,\\J,S^1)$ are derived by analysing the zeros and the symmetries of the \n\\emph{Hilbert polynomial} of $(\\mathsf{M},\\mathsf{J})$, which is defined as follows.\nLet $\\mathbb{L}_0\\to \\mathsf{M}$ be a line bundle whose first Chern class $\\mathsf{c}_1(\\mathbb{L}_0)$ is $\\eta_0=\\frac{\\mathsf{c}_1}{\\k0}$. \nThen the Hilbert polynomial $\\Hi(z)$ is the polynomial in $\\mathbb{R}[z]$ that, at integer values $k\\in {\\mathbb{Z}}$, gives the topological index of the bundle $\\mathbb{L}_0^k$, the $k$-tensor power of\n$\\mathbb{L}_0$ (note that $\\eta_0$ is only defined up to torsion, however $\\Hi(z)$ does not depend on this choice, see Sect.\\ \\ref{equations chern}).\nBy the Atiyah-Singer formula, for every $k\\in {\\mathbb{Z}}$, the integer $\\Hi(k)$ can be expressed in terms of Chern numbers of $(\\M,\\J,S^1)$:\n\\begin{equation}\\label{H and c}\n\\Hi(k)=\\left( \\sum_{h=0}^n \\frac{(k \\,\\eta_0)^h}{h!}\\right)\\ttot[\\mathsf{M}]\n= \\left( \\sum_{h=0}^n \\frac{(k \\,\\mathsf{c}_1)^h}{\\k0^h\\,h!}\\right)\\left( 1+\\frac{\\mathsf{c}_1}{2}+ \\frac{\\mathsf{c}_1^2+\\mathsf{c}_2}{12}+\\cdots\\right)[\\mathsf{M}]\n\\end{equation}\nwhere $\\ttot=\\sum_{j\\geq 0}T_j= 1+\\frac{\\mathsf{c}_1}{2}+ \\frac{\\mathsf{c}_1^2+\\mathsf{c}_2}{12}+\\cdots$ is the total Todd class of $\\mathsf{M}$, and $T_j\\in H^{2j}(\\mathsf{M};{\\mathbb{Z}})$ the Todd polynomials of $(\\mathsf{M},\\mathsf{J})$, for $j=0,\\ldots,n$, namely the polynomials in the Chern classes of $(\\mathsf{M},\\mathsf{J})$ belonging to the power series $\\frac{x}{1-e^{-x}}$. Note that in particular $\\Hi(0)=T_n[\\mathsf{M}]$, the Todd genus of $\\mathsf{M}$, which in turn is equal to $N_0$ (Proposition \\ref{properties P} (1)).\n\nUsing equivariant extensions of $\\mathbb{L}_0^k$ and localization in equivariant $K$-theory (the Atiyah-Segal formula \\eqref{AS formula}),\nit is proved that changing the orientation on $S^1$ implies the following\n `\\emph{reciprocity law}' for $\\Hi(z)$ (Propositions \\ref{symmetries} and \\ref{properties P} (2)):\n\\begin{equation}\\label{reciprocity}\n\\Hi(z)=(-1)^n \\Hi(-\\k0-z)\\,.\n\\end{equation}\nThis generalises, in the sense described in Sect.\\ \\ref{connections ehrhart}, a reciprocity law known for the Ehrhart polynomial of a reflexive polytope due to Hibi \\cite{Hibi}.\n\nThe next theorem is the key result of Section \\ref{equations chern}:\n\\begin{theorem}\\label{main theorem}\nLet $(\\mathsf{M},\\mathsf{J}, S^1)$ be an $S^1$-space. \nAssume that the index $\\k0$ of $(\\mathsf{M},\\mathsf{J})$ is greater or equal to $2$. Let $\\Hi(z)$ be the associated Hilbert polynomial and $\\deg(\\Hi)$ its degree.\nThen \\\\\n\\begin{align}\\label{H=0 even}\n & \\Hi(-1)=\\Hi(-2)=\\cdots = \\Hi(-\\k0+1)=0\\,.\n \\end{align}\n $\\;$\\\\\nMoreover, if $\\Hi(z)\\not\\equiv 0$, then \n\\begin{equation}\\label{bound k0}\n \\k0\\leq \\deg(\\Hi)+1\\leq n+1\\,.\n\\end{equation}\n\\end{theorem}\nEquations \\eqref{H and c} and \\eqref{H=0 even} suggest that studying the Chern numbers of $(\\M,\\J,S^1)$ for large values of $\\k0$ is easier.\n\nIn Sect.\\ \\ref{sec: generating fct} it is proved that, as a consequence of \\eqref{reciprocity} and \\eqref{H=0 even},\n\\emph{the number of conditions that determine the coefficients of $\\Hi(z)$ is the same for $\\k0=n+1-2k$ and $\\k0=n-2k$, for every $k\\in {\\mathbb{Z}}$ such that $0\\leq k \\leq \\frac{n-1}{2}$} (see Remark \\ref{num of cds}). \nThis follows from the fact that the generating function of $\\Hi(z)$ is\n a rational function of the form $\\Gen(t)=\\mathrm{U}(t)\/(1-t)^{\\deg(\\Hi)+1}$, where $\\mathrm{U}(t)$ is a polynomial which ---up to a power of $t$---\n is \\emph{self-reciprocal} or \\emph{palindromic} (see Proposition \\ref{gen fct hilbert} and Corollary \\ref{U palindrom}). \n\nUsing the results above we prove that, for $\\k0\\in \\{n,\\,n+1\\}$, $\\Hi(z)$ is completely determined by $N_0$; more precisely we prove that $\\Hi(z)=N_0 \\Hi_{\\overline{M}}(z)$, the manifold $\\overline{M}$ being ${\\mathbb{C}} P^n$ for $\\k0=n+1$ and the hyperquadric $Q_n$ in ${\\mathbb{C}} P^{n+1}$ for $\\k0=n$. This gives equations for the combinations of Chern numbers $\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]$ in terms of $n$, for every $h=0,\\ldots,n$, and in particular the values of $\\mathsf{c}_1^n[\\mathsf{M}]$ and $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$ (see Propositions \\ref{cor n+1} and \\ref{cor n}). \n \nWhen $\\k0=n-1$ (and $n\\geq 2$) or $\\k0=n-2$ (and $n\\geq 3$), $\\Hi(z)$ and the combinations of Chern numbers $\\mathsf{c}_1^h\\,T_{n-h}[\\mathsf{M}]$, for $h=0,\\ldots,n$, depend on a parameter. We compute explicitly their expressions in terms of this parameter (Propositions \\ref{k0=n-1} and \\ref{k0=n-2}) and determine a linear equation in $\\mathsf{c}_1^n[\\mathsf{M}]$ and $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$ which depends on $n$ and $N_0$: this is the content of Corollary \\ref{relation c122} and Corollary \\ref{relation c122 2}. \n\nInter alia, we study the position of the roots of $\\Hi(z)$ for $\\k0\\geq n-2$ and $\\k0\\neq 0$, making\nconnections with the work of Rodriguez-Villegas \\cite{RV} and Golyshev \\cite{Go}.\n \nFinally, in Section \\ref{examples} we investigate how in low dimensions the Chern numbers of $(\\mathsf{M},\\mathsf{J},S^1)$ depend on the integers $N_j$, for $j=0,\\ldots,n$, defined as the number\nof fixed points with $j$ negative weights. \nFor instance, we prove that for $\\k0=n$ or $n+1$, and $n\\leq 4$, all the Chern numbers of $(\\mathsf{M},\\mathsf{J},S^1)$ can be expressed as linear combinations of the $N_j$'s,\nand when $n=2$ having $\\k0=2$ or $3$ implies relations among the $N_j$'s. \n\n\\medskip\n\n\\subsection{Applications to symplectic manifolds}\\label{atsm} \nIn order to apply the results we obtained for almost complex manifolds to symplectic manifolds,\nlet $\\mathsf{J}\\colon T\\mathsf{M} \\to T\\mathsf{M}$ be an almost complex structure compatible with $\\omega$, namely $\\omega(\\cdot, \\mathsf{J} \\cdot)$ is a Riemannian metric. Since the set of such structures\nis contractible, we can define complex invariants of $T\\mathsf{M}$, namely Chern classes and Chern numbers. \n\nLet $(\\mathsf{M},\\omega)$ be a compact, connected symplectic manifold endowed with a symplectic circle action with isolated fixed points. \\emph{Such a space is henceforth denoted by $(\\mathsf{M},\\omega,S^1)$}. \nIt follows that the $1$-form $\\iota_{\\xi^\\#}\\omega$ is closed; here $\\xi^\\#$ denotes the vector field generated by the circle action. \nIf the $1$-form $\\iota_{\\xi^\\#}\\omega$ is \\emph{exact} the action is said to be \\emph{Hamiltonian}, otherwise we call it \\emph{non-Hamiltonian}.\nIn the first case, if $\\psi\\colon \\mathsf{M}\\to \\mathbb{R}$ is a function satisfying\n$\n\\iota_{\\xi^\\#}\\omega=-d\\psi\\,,\n$\nthen $\\psi$ is called a \\emph{moment map} for the $S^1$-action. \n\nThe first consequence of Theorem \\ref{main theorem} in the symplectic category follows from the fact that,\nif the action is Hamiltonian, $\\Hi(z)$ can never be\nidentically zero (see Remark \\ref{H Ham}), and the index coincides with the minimal Chern number (see Remark \\ref{mcn}), leading to the following\n\\begin{corollary}\\label{minimal chern ham}\nLet $(\\mathsf{M},\\omega)$ be a compact, connected symplectic manifold of dimension $2n$.\n If $(\\mathsf{M},\\omega)$ supports a Hamiltonian $S^1$-action with isolated fixed points, then its minimal Chern number coincides with the index $\\k0$, and the following inequalities hold $$1\\leq \\k0 \\leq n+1.$$\n\\end{corollary}\n This result can be considered the analogue in the Hamiltonian category of a theorem of Michelsohn \\cite[Cor.\\ 7.17]{Mi}, which asserts that the index of a compact complex manifold admitting\n a K\\\"ahler metric with positive Ricci curvature is at most $n+1$. The same conclusion also holds if $(\\mathsf{M},\\mathsf{J})$ is a compact almost complex\n manifold which can be endowed with a quasi-ample line bundle; this result is due to Hattori \\cite{Ha} and is discussed in Remark \\ref{hattori rmk}.\n If a compact symplectic manifold can be endowed with a non-Hamiltonian circle action with isolated fixed points, then\n there are three possibilities for the index and the Hilbert polynomial (see Corollary \\ref{bound on k0 s} and Remark \\ref{rmk 1}).\n \nThere are plenty of examples of compact symplectic manifolds that can be\nendowed with a Hamiltonian circle action with isolated fixed points. Until not so long ago, it was indeed believed that\nevery symplectic circle action with isolated fixed points would be\nHamiltonian.\nThis is sometimes also known as the `McDuff conjecture', and holds\\footnote{For $n=1$, the only compact symplectic surface that can be endowed\nwith a symplectic circle action with isolated fixed points is the sphere, which is simply connected, hence the action is Hamiltonian. \nFor $n=2$ the same conclusion holds by a result of McDuff in \\cite{MD1}.\nIn the same paper the author also proves the existence of a six-dimensional compact symplectic manifold with a non-Hamiltonian action, but the fixed point set is not discrete.}\n for $n=1$ and $2$ \\cite{MD1}, as well as in many other particular cases (see for instance \\cite{Fe,Fr,Go1,Go2,L,Ono,TW,J}).\nIt is only very recently that Tolman announced the following striking result:\n\\begin{thm}[Tolman '15 \\cite{T3}]\\label{tolman 6}\nThere exists a non-Hamiltonian symplectic circle action with exactly 32 fixed points on a closed, connected, six-dimensional symplectic manifold $(\\widetilde{M},\\omega)$.\n\\end{thm}\nThis theorem implies the existence of a non-Hamiltonian symplectic circle action with discrete fixed point set for every $n\\geq 3$: it is sufficient to take products $\\widetilde{M}\\times M$, where $M$ is a compact symplectic manifold endowed with a Hamiltonian circle action with $|M^{S^1}|<\\infty$\n(see also \\cite[Cor.\\ 1.2]{T3}, where $M={\\mathbb{C}} P^{n-3}$). However these products\ngive, so far, the only known examples of symplectic manifolds with non-Hamiltonian circle actions with discrete fixed point set, and the construction of new examples seems far from trivial. \nThus we ask the following `weaker' question:\n\\begin{question}\\label{q 2}\nLet $(\\mathsf{M},\\omega)$ be a compact, connected symplectic manifold. Are there topological conditions which imply that $(\\mathsf{M},\\omega)$ can only support a Hamiltonian or only a non-Hamiltonian action?\n\\end{question}\nThe answer we give to Question \\ref{q 2} is in terms of the Chern numbers of $(\\mathsf{M},\\omega,S^1)$.\nIt is already known that if $\\mathsf{c}_1$ is torsion in $H^2(\\mathsf{M};{\\mathbb{Z}})$, the manifold cannot support any Hamiltonian circle action (see \\cite[Prop.\\ 4.3]{GPS}, or also Lemma \\ref{Lemma:c1 not torsion}, and \\cite[Lemma 3.8]{T2}). \nThus the analysis we carry out to answer Question \\ref{q 2} is under the hypothesis that $\\mathsf{c}_1$ is not torsion. \nA result of Feldman \\cite{Fe} asserts that the Todd genus $T_n[\\mathsf{M}]$ of $(\\mathsf{M},\\omega,S^1)$ is either $1$ or $0$, and it is zero precisely if the action is non-Hamiltonian.\nAlthough Feldman's result is very strong and gives an answer to Question \\ref{q 2}, computing the Todd genus in high dimensions is difficult, since $T_n$ becomes a complicated combination of Chern classes. \nIn some sense, our results can be regarded as a refinement of Feldman's, since we prove that\n\\emph{given a compact symplectic manifold $(\\mathsf{M},\\omega)$, if certain combinations of Chern numbers vanish, then $(\\mathsf{M},\\omega)$ cannot support any Hamiltonian circle action with isolated fixed points}. These combinations of Chern numbers depend on $\\k0$, and are easier to compute than the Todd genus if $\\k0$ is big enough (see Corollary \\ref{cor non ham 2}). \nFor $\\k0\\geq n-2$, we strengthen the result above by giving the possible values of $\\mathsf{c}_1^n[\\mathsf{M}]$, $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$ or a combination of them, these values depending on whether the action is Hamiltonian or not.\nThis is summarized in the following\n \\begin{thm}[{\\bf Hamiltonian vs non-Hamiltonian symplectic $S^1$-actions}]\\label{nHam-char}\n$\\;$\\\\\n\\noindent\nLet $(\\mathsf{M},\\omega)$ be a compact, connected symplectic manifold, and suppose it can be endowed with a symplectic circle action with isolated fixed points.\nLet $\\k0$ be its index. Then:\n\\begin{itemize}\n\\item[(I)] If $\\k0=0$ or $\\k0 > n+1$ the action is non-Hamiltonian and $\\mathsf{c}_1^n[\\mathsf{M}]=\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=0$.\n\\item[(II)] If $\\k0=n+1$ then $(\\mathsf{c}_1^n[\\mathsf{M}],\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}])$ is equal to $\\Big((n+1)^n,\\frac{n(n+1)^{n-1}}{2}\\Big)$ or $(0,0)$. \n\\item[(III)] If $\\k0=n$ then $(\\mathsf{c}_1^n[\\mathsf{M}],\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}])$ is equal to $\\Big( 2n^n,n^{n-2}(n^2-n+2)\\Big)$ or $(0,0)$.\n\\end{itemize}\nMoreover, in \\emph{(II)} and \\emph{(III)} the\naction is Hamiltonian if and only if $\\mathsf{c}_1^n[\\mathsf{M}]\\neq 0$ (or equivalently if and only if $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]\\neq 0$).\n\\begin{itemize}\n\\item[(IV)] If $\\k0=n-1$ and $n\\geq 2$ then \n\\begin{equation}\\label{mm1}\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]-\\frac{n(n-3)}{2(n-1)^2}\\mathsf{c}_1^n[\\mathsf{M}]\\quad \\in \\Big\\{0,12 (n-1)^{n-2}\\Big\\}.\n\\end{equation}\n\\item[(V)] If $\\k0=n-2$ and $n\\geq 3$ then \n\\begin{equation}\\label{mm2}\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]-\\frac{n-3}{2(n-2)}\\mathsf{c}_1^n[\\mathsf{M}]\\quad \\in \\Big\\{0,24 (n-2)^{n-2}\\Big\\}.\n\\end{equation}\n\\end{itemize}\nMoreover, in \\emph{(IV)} (resp.\\ \\emph{(V)}) the action is Hamiltonian if and only if the combination of Chern numbers in \\eqref{mm1} (resp.\\ \\eqref{mm2}) does not vanish.\n\\end{thm}\n\\begin{rmk}\n\\begin{itemize}\n\\item[(1)] This theorem implies that for $\\k0\\geq n-2$ the Chern numbers $\\mathsf{c}_1^n[\\mathsf{M}]$ and $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$ of $(\\mathsf{M},\\omega,S^1)$ are very \\emph{rigid}. Hence it gives necessary conditions for a compact, connected symplectic manifold $(\\mathsf{M},\\omega)$ with $\\k0> \\max\\{n-3,0\\}$ to support a symplectic circle action with isolated fixed points. \n\\item[(2)] Given a compact, connected symplectic manifold $(\\mathsf{M},\\omega)$ of dimension $2n$, Theorem \\ref{nHam-char} implies that\nif the index satisfies $\\k0\\geq n$ and $\\mathsf{c}_1^n[\\mathsf{M}]$ or $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$ vanish, then $(\\mathsf{M},\\omega)$ cannot be endowed with \\emph{any} Hamiltonian circle\naction with isolated fixed points. A similar conclusion holds for $\\k0\\in \\{n-2,n-1\\}$, by considering the combinations of Chern numbers in \\eqref{mm1} and \\eqref{mm2}.\n\\item[(3)] The above results are stated in terms of the Chern numbers $ \\mathsf{c}_1^n[\\mathsf{M}]$ and $ \\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$; however similar conclusions \ncan be obtained for $\\mathsf{c}_1^h\\,T_{n-h}[\\mathsf{M}]$, for $h=0,\\ldots,n$ (see Remark \\ref{other comb}).\n\\end{itemize}\n\\end{rmk}\nFinally, in Section \\ref{examples} we analyse the geography problem for $(\\mathsf{M},\\omega,S^1)$ when $n\\leq 4$.\nOne of the goals is to find,\nin the Hamiltonian case,\nformulas for the Chern numbers in terms of $\\k0$ and the Betti numbers of $\\mathsf{M}$. \nFor instance, the geography problem for $n=2$ can be completely solved (Corollary \\ref{geo s}), and for $n=3,4$\nwe solve it for every $\\k0\\geq n$ (Propositions \\ref{dim 6} and \\ref{dim 8}).\nAs a byproduct of the investigation in dimension $8$, we prove that if a compact, connected symplectic manifold\nof dimension $8$ supports a Hamiltonian $S^1$-action with\nisolated fixed points, and if the minimal Chern number is even, then $\\mathsf{c}_2^2[\\mathsf{M}]+2\\, b_2(\\mathsf{M})=98 +b_4(\\mathsf{M})$ (Corollary \\ref{c228h}). \n\n\n\\vspace{.5cm} \n\\textbf{Acknowledgements.}\\\nFirst of all, I would like to thank Leonor Godinho for many fruitful conversations during the time I spent at Instituto Superior T\\'ecnico, for inspiring this work and reading previous drafts. I would also like to thank Hansj\\\"org Geiges for useful discussions, and in particular for suggesting Remark \\ref{mcn}. Frederik von Heymann explained to me many useful facts about reflexive polytopes, and strongly inspired Section \n\\ref{connections ehrhart}.\n\nAlthough I have never met him, I would like to dedicate this work to the memory of Akio Hattori who, through his articles, taught me so much. \n\n\\section{Background and preliminary results}\\label{background}\nThe main purpose of this section is to recall background material, set up notation and state preliminary results needed in the forthcoming sections.\n\nLet $(\\mathsf{M},\\mathsf{J})$ be a compact, connected almost complex manifold of dimension $2n$.\nThus $\\mathsf{J}\\colon T\\mathsf{M} \\to T\\mathsf{M}$ is a complex structure on the tangent bundle of $\\mathsf{M}$, and \nfor such manifold we consider the Chern classes of the tangent bundle, denoted by $\\mathsf{c}_j\\in H^{2j}(\\mathsf{M};{\\mathbb{Z}})$\\footnote{To avoid confusion, if in the same paragraph we also deal with\nChern classes of other bundles, we\nwill denote the Chern \nclasses of the tangent bundle by $\\mathsf{c}_j(\\mathsf{M})$.},\nas well as the Chern numbers \n$ \\mathsf{c}_{j_1}\\cdots \\mathsf{c}_{j_l}[\\mathsf{M}]\\in {\\mathbb{Z}}$, for every partition $(j_1,\\ldots,j_l)$ of $n$, i.e.\\ $j_1+\\ \\cdots \\ +j_l=n$ and $j_m\\in \\mathbb{N}$ for $m=1\\ldots,l$.\n\nMoreover assume that $(\\M,\\J,S^1)$ is an $S^1$-space, i.e.\\ $(\\mathsf{M},\\mathsf{J})$ is endowed with a $\\mathsf{J}$-preserving $S^1$-action with nonempty and discrete fixed point set $\\mathsf{M}^{S^1}=\\{p_0,\\ldots,p_N\\}$, for some $N\\in {\\mathbb{Z}}_{>0}$.\n\n\nFor every $p_i\\in M^{S^1}$ we denote by $w_{i,1},\\ldots,w_{i,n}$ the \\emph{weights} of the (isotropy) action of $S^1$ at $p_i$, i.e.\\\nthe $S^1$ representation induced on $T_p\\mathsf{M}$ is given by\n\\begin{equation}\\label{weights def}\n\\alpha\\cdot(z_1,\\ldots,z_n)=(\\alpha^{w_{i,1}}z_1,\\ldots,\\alpha^{w_{i,n}}z_n)\\;\\quad\\mbox{for every}\\quad \\alpha\\in S^1,\n\\end{equation}\nfor a suitable choice of complex coordinates $(z_1,\\ldots,z_n)$ on $T_p\\mathsf{M}\\simeq {\\mathbb{C}}^n$. We also denote by $W_i$ the (multi)set of weights at $p_i$, i.e.\\;$W_i=\\{w_{i,1},\\ldots,w_{i,n}\\}$.\nNote that $w_{i,j}$ is nonzero for every $i=1\\ldots,N$ and $j=1\\ldots,n$, since the isotropy action commutes with the action on the manifold $\\mathsf{M}$,\nand $\\mathsf{M}^{S^1}$ is discrete.\nFinally, we denote by $\\lambda_i$ the number of negative weights at $p_i\\in M^{S^1}$ and by $N_j$ the number of fixed points with exactly $j$ negative weights, for every $j=0,\\ldots,n$. From \\cite[Proposition 2.6]{Ha} we have that\n\\begin{equation}\\label{NiN}\nN_j=N_{n-j}\\quad \\mbox{for every}\\quad j=0,\\ldots,n\\,. \n\\end{equation}\n\n\nLet $K(\\mathsf{M})$ (resp.\\;$K_{S^1}(\\mathsf{M})$) be the ordinary (resp.\\;$S^1$-equivariant) $K$-theory ring of $\\mathsf{M}$, i.e.\\;the abelian group associated to the\nsemigroup of isomorphism classes of complex vector bundles (resp.\\;complex $S^1$-vector bundles) over $\\mathsf{M}$,\nendowed with the direct sum $\\oplus$ and tensor product $\\otimes$ operation.\nThus in particular\n$K(\\{pt\\})\\simeq {\\mathbb{Z}}$ and $K_{S^1}(\\{pt\\}) \\simeq R(S^1),$\nthe character ring of $S^1$. Henceforth, we identify the latter with the Laurent\npolynomial ring ${\\mathbb{Z}}[t,t^{-1}]$, where $t$ denotes the standard $S^1$-representation. \n\nLet $H_{S^1}^*(\\mathsf{M};{\\mathbb{Z}})$ be the $S^1$-equivariant cohomology of $\\mathsf{M}$ with ${\\mathbb{Z}}$ coefficients; we recall that this is defined to be the\nordinary cohomology of the Borel model, i.e.\\;$\nH_{S^1}^*(\\mathsf{M};{\\mathbb{Z}}):=H^*(\\mathsf{M}\\times_{S^1}S^{\\infty};{\\mathbb{Z}})\\,,\n$ \nwhere $S^{\\infty}$ is the unit sphere in ${\\mathbb{C}}^{\\infty}$. Thus in particular $H_{S^1}^*(\\{pt\\};{\\mathbb{Z}})={\\mathbb{Z}}[x]$, where $x$ has degree $2$.\n\nFinally, let $\\pic(\\mathsf{M})$ (resp.\\;$\\pic_{S^1}(\\mathsf{M})$) be the Picard group of isomorphism classes of complex line bundles (resp.\\;equivariant complex line bundles) over $\\mathsf{M}$.\n\nIn the rest of the section, $\\mathcal{H}(\\cdot)$ (resp.\\;$\\mathcal{H}_{S^1}(\\cdot)$) will either denote the cohomology (resp.\\;equivariant cohomology) ring \nwith ${\\mathbb{Z}}$ coefficients, the $K$-theory (resp.\\;equivariant $K$-theory) ring, or the Picard (resp.\\;equivariant Picard) group.\n\nFor $p\\in \\mathsf{M}^{S^1}$ let $i_p\\colon \\{p\\}\\hookrightarrow \\mathsf{M}$ and $i\\colon \\mathsf{M}^{S^1}\\hookrightarrow \\mathsf{M}$ denote the natural inclusions; since they are equivariant we have the following induced maps:\n$$\ni_p^*\\colon \\mathcal{H}_{S^1}(\\mathsf{M})\\to \\mathcal{H}_{S^1}(\\{p\\})\n$$\nand\n\\begin{equation}\\label{istar}\n i^*=\\bigoplus_{p\\in \\mathsf{M}^{S^1}}i_p^*\\colon \\mathcal{H}_{S^1}(\\mathsf{M})\\to \\mathcal{H}_{S^1}(\\mathsf{M}^{S^1})=\\bigoplus_{p\\in \\mathsf{M}^{S^1}}\\mathcal{H}_{S^1}(\\{p\\})\\;.\n\\end{equation}\nWe denote $i_p^*(K)$ simply by $K(p)$, for every $p\\in \\mathsf{M}^{S^1}$ and $K\\in \\mathcal{H}_{S^1}(\\mathsf{M})$.\n\nObserve that the unique map\n$ \\mathsf{M}\\to \\{pt\\}$ induces maps \n\\begin{equation*}\n\\mathcal{H}_{S^1}(\\{pt\\})\\to \\mathcal{H}_{S^1}(\\mathsf{M})\\quad\\text{and} \\quad \\mathcal{H}(\\{pt\\})\\to \\mathcal{H}(\\mathsf{M}),\n\\end{equation*}\nwhich give $\\mathcal{H}_{S^1}(\\mathsf{M})$ the structure of an $\\mathcal{H}_{S^1}(\\{pt\\})$-module, and $\\mathcal{H}(\\mathsf{M})$ the structure of an $\\mathcal{H}(\\{pt\\})$-module.\n\nFinally, if $e$ denotes the identity element in $S^1$, the inclusion homomorphism \n$\\{e\\}\\hookrightarrow S^1$ induces a restriction map, also called the ``forgetful homomorphism\" \n\\begin{equation}\\label{restriction}\n r_{\\mathcal{H}}\\colon \\mathcal{H}_{S^1}(\\mathsf{M})\\to \\mathcal{H}(\\mathsf{M})\\;.\n\\end{equation}\nWhen $M$ is a point, $r_{\\mathcal{H}}$ coincides with the evaluation at $x=0$ in cohomology, and \n with the evaluation at $t=1$ in $K$-theory and in the Picard group. \n The homomorphism \\eqref{restriction} will be denoted by $r_H$ in cohomology, by $r_K$ in $K$-theory and by $r_{\\pic}$ for the Picard group. \n\n\\subsection{Indices of $K$-theory classes}\\label{subsec: indeces}\nLet \n\\begin{equation}\\label{indK}\n \\ind\\colon K(\\mathsf{M})\\to K(pt)\\simeq{\\mathbb{Z}}\n\\end{equation}\n and \n \\begin{equation}\\label{indKe}\n \\ind_{S^1}\\colon K_{S^1}(\\mathsf{M})\\to K_{S^1}(pt)\\simeq {\\mathbb{Z}}[t,t^{-1}] \n \\end{equation}\n be the index homomorphisms (or $K$-theoretic push forwards) in ordinary and equivariant $K$-theory.\nBy the Atiyah-Singer formula, the index in \\eqref{indK} \ncan be computed as \n\\begin{equation}\\label{AT formula}\n\\ind(V)= \\ch(V)\\ttot [\\mathsf{M}]\\;,\\quad\\mbox{for every}\\quad V\\in K(\\mathsf{M}),\n\\end{equation}\nwhere $\\ch(\\cdot)$ is the Chern character homomorphism $\\ch\\colon K(\\mathsf{M})\\to H^*(\\mathsf{M};\\mathbb{Q})$, and $\\ttot$ is the total Todd class of $\\mathsf{M}$, i.e.\\;the cohomology \nclass in $H^*(M;{\\mathbb{Z}})$ associated to the power series $\\displaystyle\\frac{x}{1-e^{-x}}$. This is a rational combination of Chern classes, and\nthe first terms of $\\ttot$ are given by\n\\begin{equation}\\label{Todd}\n \\ttot=\\sum_{j\\geq 0}T_j=1+\\frac{\\mathsf{c}_1}{2}+\\frac{\\mathsf{c}_1^2+\\mathsf{c}_2}{12}+\\frac{\\mathsf{c}_1\\mathsf{c}_2}{24}+\\frac{-\\mathsf{c}_1^4+4\\mathsf{c}_1^2\\mathsf{c}_2+3\\mathsf{c}_2^2+\\mathsf{c}_1\\mathsf{c}_3-\\mathsf{c}_4}{720}+\\ldots \n\\end{equation}\nwhere $T_j\\in H^{2j}(\\mathsf{M};{\\mathbb{Z}})$ for every $j$. We also recall that the Todd genus $\\td(\\mathsf{M})$ of $\\mathsf{M}$ is given by\n$$\n\\td(\\mathsf{M})= \\ttot[\\mathsf{M}]=T_n[\\mathsf{M}]\\;.\n$$\n\n\nBy the Atiyah-Segal formula \\cite{AS}, \nthe equivariant index \\eqref{indKe} of a class $V\\in K_{S^1}(\\mathsf{M})$ can be computed \nin terms of $i^*(V)$ and the $S^1$ isotropy representation on $T\\mathsf{M}\\rvert_{\\mathsf{M}^{S^1}}$. Since $M^{S^1}$ is discrete, the Atiyah-Segal formula in this case gives \n\\begin{equation}\\label{AS formula}\n\\ind_{S^1}(V)=\\sum_{i=0}^N \\frac{V(p_i)}{\\prod_{j=1}^n (1-t^{-w_{i,j}})}\\,,\\quad\\mbox{for every}\\quad V\\in K_{S^1}(\\mathsf{M})\\;.\n\\end{equation} \nBy \\eqref{AT formula}, \\eqref{AS formula} and the commutativity of the following diagram \n\\begin{equation}\\label{K commutes}\n\\xymatrix{\nK_{S^1}(\\mathsf{M}) \\ar[r]^{r_K} \\ar[d]_{\\ind_{S^1}} & K(\\mathsf{M}) \\ar[d]_{\\ind} \\\\\n {\\mathbb{Z}}[t,t^{-1}] \\ar[r]^{r_K} & {\\mathbb{Z}}.\n } \\\n\\end{equation}\nit follows that for every $V\\in K_{S^1}(M)$ we have\n\\begin{equation}\\label{formula index 2}\n\\left(\\sum_{i=0}^N \\frac{V(p_i)}{\\prod_{j=1}^n (1-t^{-w_{i,j}})}\\right)_{\\rvert_{t=1}}=\nr_K(\\ind_{S^1}(V))=\\ind(r_K(V))=\n \\ch(r_K(V))\\ttot[\\mathsf{M}]\\;.\n\\end{equation}\n\nWe conclude this subsection by recalling the Atiyah-Bott-Berline-Vergne Localization formula \\cite{At,BV}:\n\\begin{theorem}[ABBV Localization formula]\\label{abbv formula}\nLet $\\mathsf{M}$ be a compact oriented manifold endowed with a smooth $S^1$-action.\nGiven $\\mu\\in H_{S^1}^*(\\mathsf{M};\\mathbb{Q})$\n\\begin{equation*}\n\\mu[\\mathsf{M}]= \\sum_{F}\\frac{i_F^*(\\mu)}{e^{S^1}(N_F)}[F]\\;,\n\\end{equation*}\nwhere the sum is over all the fixed-point set components $F$ of the action, and\n$e^{S^1}(N_F)$ is the equivariant Euler class of the normal bundle to $F$.\n\\end{theorem} \n\n\\subsection{Equivariant Chern classes and equivariant complex line bundles}\\label{ecc}\n\nGiven a complex vector bundle $V\\to \\mathsf{M}$, denote by\n$\\mathsf{c}(V)=\\sum_i\\mathsf{c}_i(V)\\in H^*(\\mathsf{M};{\\mathbb{Z}})$ the total Chern class of $V$, and if $V$ is equivariant, by $\\mathsf{c}^{S^1}(V)=\\sum_i\\mathsf{c}_i^{S^1}(V)\\in H^*_{S^1}(\\mathsf{M};{\\mathbb{Z}})$\nthe total equivariant Chern class, i.e.\\;the total Chern class of the bundle\n$V\\times_{S^1}S^{\\infty}\\to \\mathsf{M}\\times_{S^1}S^{\\infty}$.\nIt is easy to check that when $V=T\\mathsf{M}$, if $\\mathsf{c}^{S^1}(\\mathsf{M})$ denotes the total equivariant Chern class \nof the tangent bundle $T\\mathsf{M}$, then \nfor every $p_i\\in M^{S^1}$, $\\mathsf{c}^{S^1}(\\mathsf{M})(p_i)=\\prod_{j=1}^n(1+w_{i,j}x)$, and hence\n$\\mathsf{c}^{S^1}_j(\\mathsf{M})(p_i)=\\sigma_j(w_{i,1},\\ldots,w_{i,n})x^j$, where $\\sigma_j(x_1,\\ldots,x_n)$ denotes the $j$-th elementary polynomial in $x_1,\\ldots,x_n$.\n\nIf $(\\mathsf{M},\\mathsf{J})$ is acted on by a circle $S^1$ preserving the almost complex structure,\nit is a natural question to ask whether a given complex vector bundle $V$ over $\\mathsf{M}$ admits an equivariant extension, i.e.\\;whether the $S^1$-action can be\nlifted to $V$, making the projection $V\\to \\mathsf{M}$ equivariant. \nThis question has been studied in different settings, and \nfor (complex) line bundles $\\mathbb{L}$ it has been completely answered by Hattori and Yoshida \\cite[Theorem 1.1, Corollary 1.2]{HY} (see also \\cite{HL,Mu} and \\cite[Appendix C]{GKS}); \nhere we summarise their main result in a different language.\n\\begin{theorem}[Hattori-Yoshida]\nThe equivariant first Chern class\n\\begin{equation}\\label{isom ce}\n\\mathsf{c}_{1}^{S^1}\\colon \\pic_{S^1}(\\mathsf{M})\\to H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})\n\\end{equation} \nis an isomorphism. As a consequence, \na line bundle $\\mathbb{L}$ admits an equivariant extension if and only if its first Chern class $\\mathsf{c}_{1}^{S^1}(\\mathbb{L})$ is in the image\nof the restriction map\n\n\\begin{equation}\\label{restriction H2}\nr_H\\colon H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})\\to H^2(\\mathsf{M};{\\mathbb{Z}}). \n\\end{equation}\n\\end{theorem}\n\nThe second assertion follows from the commutativity of the following diagram\n$$\n \\xymatrix{ \n\\pic_{S^1}(\\mathsf{M}) \\ar[d]_{r_{\\pic}} \\ar[r]^-{\\mathsf{c}_{1}^{S^1}} & H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})\\ar[d]^{r_H} \\\\\n\\pic(\\mathsf{M}) \\ar[r]^-{\\mathsf{c}_1} & H^2(\\mathsf{M};{\\mathbb{Z}}) \\\\\n}\n$$\nand the fact that the first Chern class map $\\mathsf{c}_1$ on the bottom row is an isomorphism.\n\nMoreover, for any line bundle $\\mathbb{L}$ whose first Chern class is in the image of \\eqref{restriction H2}, which will henceforth be called\n\\emph{admissible},\nall the possible equivariant\nextensions are parametrised by $H^2({\\mathbb{C}} P^{\\infty};{\\mathbb{Z}})\\simeq {\\mathbb{Z}}$. More precisely, given an admissible $\\mathbb{L}$ and two equivariant\nextensions $\\mathbb{L}^{S^1}_1$ and $\\mathbb{L}^{S^1}_2$, there exists $a\\in {\\mathbb{Z}}$ such that $\\mathsf{c}_{1}^{S^1}(\\mathbb{L}^{S^1}_1)-\\mathsf{c}_{1}^{S^1}(\\mathbb{L}^{S^1}_2)=ax$. In particular we have that \n\\begin{equation}\\label{trivial constant}\n\\mbox{\\emph{if}}\\;\\;\\mathbb{L} \\;\\;\\mbox{\\emph{is trivial, then }}\\mathsf{c}_{1}^{S^1}(\\mathbb{L}^{S^1})(p)=ax\\quad\\mbox{\\emph{for every} }p\\in \\mathsf{M}^{S^1},\\mbox{ \\emph{for some} }a\\in {\\mathbb{Z}}.\n\\end{equation}\n\n\nIn \\cite[Lemma 3.2]{Ha}, Hattori proves that if $\\mathbb{L}$ is admissible, and $\\mathbb{L}'$ is such that $\\mathsf{c}_1(\\mathbb{L})=k\\mathsf{c}_1(\\mathbb{L}')$ for some nonzero integer $k$,\nthen $\\mathbb{L}'$ is also admissible; moreover every line bundle whose first Chern class is in $\\tor(H^2(\\mathsf{M};{\\mathbb{Z}}))$, the torsion subgroup of $H^2(\\mathsf{M};{\\mathbb{Z}})$, is admissible.\nAn example of admissible line bundle is given by the determinant line bundle $\\Lambda^n(T\\mathsf{M})$.\nIn fact it is well-known that $\\mathsf{c}_1(\\mathsf{M})$ always admits an equivariant extension, given by the equivariant\nfirst Chern class $\\mathsf{c}_{1}^{S^1}(\\mathsf{M})$. Hence $\\Lambda^n(T\\mathsf{M})$ is admissible, since $\\mathsf{c}_1(\\Lambda^n(T\\mathsf{M}))=\\mathsf{c}_1(\\mathsf{M})$.\nMoreover the trivial bundle is clearly admissible.\n\nLet $\\mathcal{L}$ be the lattice given by $H^2(\\mathsf{M};{\\mathbb{Z}})\/\\tor(H^2(\\mathsf{M};{\\mathbb{Z}}))$ and $$\\pi\\colon H^2(\\mathsf{M};{\\mathbb{Z}})\\to \\mathcal{L}$$ the projection. The following lemma is an immediate consequence of \\cite[Lemma 3.2]{Ha}.\n\\begin{lemma}\\label{line admissible}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space and let $\\mathsf{c}_1$ be the first Chern class of the tangent bundle. \nSuppose that $\\mathsf{c}_1$ is not a torsion element, i.e.\\;$\\pi(\\mathsf{c}_1)\\neq 0$, and let $\\eta$ be a primitive element in $\\mathcal{L}$ such that $\\pi(\\mathsf{c}_1)=k_0\\eta$, for some $k_0\\in {\\mathbb{Z}}\\setminus\\{0\\}$. Then every line bundle $\\mathbb{L}$ \nsuch that $\\pi(\\mathsf{c}_1(\\mathbb{L}))=k\\,\\eta$ is admissible, for every $k\\in {\\mathbb{Z}}$.\n\\end{lemma}\n\nObserve that the {\\bf index} $\\k0$ of $(\\mathsf{M},\\mathsf{J})$, as defined in the introduction, is the same as the largest integer satisfying $\\pi(\\mathsf{c}_1)=\\k0 \\pi(\\eta_0)$, for some non-torsion $\\eta_0\\in H^2(\\mathsf{M};{\\mathbb{Z}})$. Note that, when $\\mathsf{c}_1$ is not torsion, $\\pi(\\eta_0)$ is necessarily primitive in $\\mathcal{L}$.\n\nIn the rest of this note, we will make use of the following {\\bf convention}:\nLet $\\tau$ be an element of $H_{S^1}^2(\\mathsf{M}^{S^1};{\\mathbb{Z}})$; thus $\\tau(p)=a_px\\in H_{S^1}^2(\\{p\\};{\\mathbb{Z}})$, where $a_p\\in {\\mathbb{Z}}$ and $x$ is the generator of $H_{S^1}^2(\\{p\\};{\\mathbb{Z}})=H^2({\\mathbb{C}} P^{\\infty};{\\mathbb{Z}})$.\nFor the sake of simplicity, \\emph{we henceforth identify $\\tau\\in H_{S^1}^2(\\mathsf{M}^{S^1};{\\mathbb{Z}})$ with the map from $M^{S^1}$ to ${\\mathbb{Z}}$ which assigns to $p$ the integer $a_p$.}\n$\\;$\\\\\n\nNote that for every $\\mathbb{L}^{S^1}\\in \\pic_{S^1}(\\mathsf{M})$ and every $p_i\\in M^{S^1}$ \n\\begin{equation}\\label{elb}\n\\mathbb{L}^{S^1}(p_i)=t^{a_i},\\quad \\mbox{where}\\;\\; a_i \\;\\; \\mbox{is the integer given by}\\;\\;\\; \\mathsf{c}_{1}^{S^1}(\\mathbb{L}^{S^1})(p_i).\n\\end{equation}%\n\nIn virtue of the isomorphism \\eqref{isom ce}, given a class $\\tau\\in H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ (resp.\\;$\\tau'\\in H^2(\\mathsf{M};{\\mathbb{Z}})$), we will denote by $\\e{\\tau}$ the isomorphism class of equivariant line\nbundles whose first equivariant Chern class is $\\tau$ (resp.\\;the isomorphism class of line\nbundles whose first Chern class is $\\tau'$).\nWe conclude this section with the following\n\\begin{prop}\\label{symmetries}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space with\n$\\mathsf{M}^{S^1}=\\{p_0,\\ldots,p_N\\}$. Let $\\mathsf{c}_1$ and $\\mathsf{c}_{1}^{S^1}$ be respectively the first Chern class and the equivariant first Chern class of the tangent bundle\nof $\\mathsf{M}$. Then, for every $\\tau\\in H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ we have\n\\begin{equation}\\label{eq index symmetry}\n\\ind_{S^1}(\\e{\\tau})=(-1)^n\\ind_{\\widetilde{S}^1}(\\e{(-\\tau-\\mathsf{c}_1^{\\widetilde{S}^1})})\\,,\n\\end{equation}\nwhere $\\widetilde{S}^1$ is the circle $S^1$ with orientation reversed. \nThus\n\\begin{equation}\\label{index symmetry}\n\\ind(\\e{r_H(\\tau)})=(-1)^n\\ind(\\e{(-r_H(\\tau)-\\mathsf{c}_1)}).\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nBy \\eqref{AS formula} and \\eqref{elb} we have that\n$$\n\\ind_{S^1}(\\e{\\tau})=\\sum_{i=0}^N \\frac{t^{\\tau(p_i)}}{\\prod_{j=1}^n(1-t^{-w_{i,j}})}=\\sum_{i=0}^N\\frac{(-1)^n\\,t^{\\tau(p_i)+w_{i,1}+\\ldots+w_{i,n}}}{\\prod_{j=1}^n(1-t^{w_{i,j}})}=(-1)^n\\ind_{\\widetilde{S}^1}(\\e{(-\\tau-\\mathsf{c}_1^{\\widetilde{S}^1})})\\,,\n$$\nand \\eqref{index symmetry} follows from \\eqref{K commutes}, \\eqref{eq index symmetry} and the fact that $r_H(\\mathsf{c}_{1}^{S^1})=r_H(\\mathsf{c}_1^{\\widetilde{S}^1})=\\mathsf{c}_1$.\n\\end{proof}\n\n\n\n\\section{Computation of equivariant indices}\\label{cei}\n\nIn this section we analyse some properties of the equivariant index\nof an equivariant line bundle $\\mathbb{L}^{S^1}$. In particular we study \nunder which conditions $\\mathbb{L}^{S^1}$ is `\\emph{rigid}', namely when its equivariant index $\\ind_{S^1}(\\mathbb{L}^{S^1})$ is \n$S^1$-invariant, i.e.\\ it belongs to ${\\mathbb{Z}}\\subset {\\mathbb{Z}}[t,t^{-1}]$, and determine what the constant is in terms of the restriction to the fixed points of its equivariant first Chern class: this is the\ncontent of Theorem \\ref{trick}. As a consequence, we derive conditions that ensure the equivariant index of an equivariant line bundle to be zero. \nThis is a generalisation of arguments which had\nalready been used in different ways by several authors, see for example Hattori \\cite[Proposition 2.6]{Ha}, Hirzebruch et al.\\;\\cite[Section 5.7]{Hi}, Li \\cite{L} and \nLi-Liu \\cite[Proposition 2.5]{LL}.\n\nThe rest of the section is devoted to deriving applications of Theorem \\ref{trick} which will be used in the forthcoming sections.\n\nFor every point $p_i\\in M^{S^1}$, we order the isotropy weights $w_{i,1},\\ldots,w_{i,n}$ at $p_i$ in such a way that the first $\\lambda_{i}$ are exactly the negative weights at $p_i$.\nWe define $\\cc_1^+$ and $\\cc_1^-$ in $H_{S^1}^2(\\mathsf{M}^{S^1};{\\mathbb{Z}})$ to be\n\\begin{equation}\\label{cpcm}\n\\cc_1^+(p_i)=w_{i,\\lambda_{i}+1}+\\cdots +w_{i,n}\\;\\;\\;\\quad \\mbox{and} \\quad \\;\\;\\;\\cc_1^-(p_i)=-(w_{i,1}+\\cdots+w_{i,\\lambda_i})\\,. \n\\end{equation}\nFrom the definition it follows that $\\cc_1^+(p_i)\\geq 0$ (resp.\\;$\\cc_1^-(p_i)\\geq 0$) and equality holds if and only if $\\lambda_i=n$ (resp.\\;$\\lambda_i=0$).\nMoreover, if $\\mathsf{c}_{1}^{S^1}$ denotes the equivariant first Chern class of $\\mathsf{M}$, we have that $i^*(\\mathsf{c}_{1}^{S^1})=\\cc_1^+-\\cc_1^-$. \n\n\\begin{defin}\nA class $\\tau\\in H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ is said to be \\emph{dominated} by $\\cc_1^+$ (resp.\\;by $\\cc_1^-$) if $\\tau(p)\\leq \\cc_1^+(p)$ for every $p\\in \\mathsf{M}^{S^1}$ \n(resp.\\;if $-\\tau(p)\\leq \\cc_1^-(p)$ for every $p\\in \\mathsf{M}^{S^1}$). \n\\end{defin}\n\\begin{rmk}\\label{ex 0 and c1}\nIt is easy to check that the classes $\\mathbf{0}$ and $\\mathsf{c}_{1}^{S^1}$ are always dominated by both $\\cc_1^+$ and $\\cc_1^-$. \nMoreover, if $\\tau\\in H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ satisfies $\\tau(p)\\leq 0$ (resp.\\;$\\tau(p)\\geq 0$) for every $p\\in \\mathsf{M}^{S^1}$ then $\\tau$ is dominated by $\\cc_1^+$ (resp.\\;$\\cc_1^-$). \n\\end{rmk}\n\\begin{theorem}\\label{trick}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space with\n$\\mathsf{M}^{S^1}=\\{p_0,\\ldots,p_N\\}$. Let $\\tau$ be an element of $H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ and $\\cc_1^+$, $\\cc_1^-$ defined as above. \nFor every $p\\in \\mathsf{M}^{S^1}$, define $\\delta^+(p)$ (resp.\\;$\\delta^-(p)$) to be $1$ if $\\tau(p)=\\cc_1^+(p)$ (resp.\\;$-\\tau(p)=\\cc_1^-(p)$) and zero otherwise.\nThen\n\\begin{itemize}\n \\item[(i)] If $\\tau\\in H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ is dominated by $\\cc_1^+$ then \n $$\n \\ind_{S^1}(\\e{(-\\tau)})=\\sum_{j\\geq 0}b_jt^j\\in {\\mathbb{Z}}[t],\\quad \\mbox{and}\\quad b_0= \\sum_{i=0}^N\\delta^+(p_i)(-1)^{n-\\lambda_i}\n $$\n \\item[(ii)] If $\\tau\\in H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ is dominated by $\\cc_1^-$ then \n $$\n \\ind_{S^1}(\\e{(-\\tau)})=\\sum_{j\\leq 0}b_jt^j\\in {\\mathbb{Z}}[t^{-1}],\\quad \\mbox{and}\\quad b_0= \\sum_{i=0}^N\\delta^-(p_i)(-1)^{\\lambda_i}\n $$\n \\item[(iii)] If $\\tau\\in H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ is dominated by $\\cc_1^+$ and $\\cc_1^-$ then \n \\begin{equation}\\label{index integer}\n \\ind_{S^1}(\\e{(-\\tau)})=b_0\\in {\\mathbb{Z}}\n \\end{equation}\nwhere \n \\begin{equation}\\label{index precise}\nb_0=\\sum_{i=0}^N\\delta^+(p_i)(-1)^{n-\\lambda_i}= \\sum_{i=0}^N\\delta^-(p_i)(-1)^{\\lambda_i}.\n \\end{equation}\n\\end{itemize}\n\n\\end{theorem}\n\\begin{proof}\nBy \\eqref{AS formula} and \\eqref{elb}, we have that \n \\begin{equation}\\label{ei1}\n \\ind_{S^1}(\\e{(-\\tau)}) =\\sum_{i=0}^N \\frac{ t^{-\\tau(p_i)} }{\\prod_{j=1}^n(1-t^{-w_{i,j}}) }\n\\end{equation} \nFor every $i=0,\\ldots,N$, let $f_i(t)$ be the rational function $\\displaystyle \\frac{ t^{-\\tau(p_i)} }{\\prod_{j=1}^n(1-t^{-w_{i,j}}) }$, and observe that\n$\\sum_{i=0}^Nf_i(t)\\in {\\mathbb{Z}}[t,t^{-1}]$.\nThus, in order to prove (i), it is sufficient to prove that $\\lim_{t\\to 0}\\sum_{i=0}^Nf_i(t)$ is finite, and its value will be equal to $b_0$. Observe that by definition\nof $\\cc_1^+$, $f_i(t)$ can be rewritten as $\\displaystyle \\frac{(-1)^{n-\\lambda_i}\\;t^{-\\tau(p_i)+\\cc_1^+(p_i)}}{\\prod_{j=1}^n(1-t^{|w_{i,j}|})}$.\nSince by assumption $i^*(\\tau)$ is dominated by $\\cc_1^+$, $\\lim_{t \\to 0}f_i(t)$ is finite for all $i=0,\\ldots,N$, and by definition of $\\delta^+$ it follows that \nits value equals to $\\delta^+(p_i)(-1)^{n-\\lambda_i}$, thus proving (i).\n\nThe proof of (ii) follows by a similar argument, by taking $\\lim_{t\\to \\infty}\\sum_{i=0}^Nf_i(t)$, and by observing that $f_i(t)$ can be written as\n$ \\displaystyle \\frac{(-1)^{\\lambda_i}\\;t^{-\\tau(p_i)-\\cc_1^-(p_i)}}{\\prod_{j=1}^n(1-t^{-|w_{i,j}|})}$.\n\nFinally, (iii) follows from (i) and (ii).\n\\end{proof}\n\n\\begin{exm}\\label{exm:CP3}\nConsider $({\\mathbb{C}} P^3,\\mathsf{J})$ with the standard (almost) complex structure, and $S^1$-action given by \n$$\n\\lambda \\cdot [z_0:z_1:z_2:z_3]=[z_0:\\lambda^a z_1:\\lambda^{a+b}z_2:\\lambda^{a+b+c}z_3],\n$$\nwhere $a,b,c$ are pairwise coprime positive integers. This action is ``standard'', in the sense that it is the\nrestriction to a subtorus of dimension $1$ of the standard toric action of the $3$-dimensional torus $\\mathbb{T}^3$ on ${\\mathbb{C}} P^3$.\nThe fixed point set is given by four points $p_0,p_1,p_2,p_3$, corresponding respectively to $[1:0:0:0],[0:1:0:0],[0:0:1:0],[0:0:0:1]$. \nLet $\\tau_0$ be the generator of $H^2({\\mathbb{C}} P^3,{\\mathbb{Z}})$ such that $\\mathsf{c}_1({\\mathbb{C}} P^3)=4\\, \\tau_0$. It can be checked that $\\tau_0$ admits an equivariant\nextension\\footnote{Indeed, in this case, every class $\\gamma\\in H^j({\\mathbb{C}} P^3,{\\mathbb{Z}})$ admits an equivariant extension, for every $j$. This is due to the fact\nthat ${\\mathbb{C}} P^3$ with the above $S^1$-action is \\emph{equivariantly formal} (see for example \\cite{Ki}).} $\\tau\\in H^2_{S^1}({\\mathbb{C}} P^3,{\\mathbb{Z}})$, i.e.\\;$r_H(\\tau)=\\tau_0$; we pick\n$\\tau$ so that $\\tau(p_0)=0$. \nThe (multi)sets of isotropy weights at each fixed point, as well as $i^*(\\tau)$, $\\cc_1^+$ and $\\cc_1^-$, are given in the following table:\n\\begin{center}\n\\begin{tabular}{|l|| l|l|l|l|}\n\\hline\n & $\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;W_i$ & $\\;\\;\\;\\;\\;i^*(\\tau)$ & $\\;\\;\\;\\;\\;\\;\\;\\cc_1^+$ & $\\;\\;\\;\\;\\;\\;\\;\\cc_1^-$ \\\\ \\hline \n$p_0:$ & $\\{a,a+b,a+b+c\\}$ & $0$ & $3a+2b+c$ & $0$ \\\\ \\hline\n$p_1:$ & $\\{-a,b,b+c\\}$ & $-a$ & $2b+c$ & $a$ \\\\ \\hline \n$p_2:$ & $\\{-b,-a-b,c\\}$ & $-a-b$ & $c$ & $a+2b$ \\\\ \\hline \n$p_3$ & $\\{-c,-b-c,-a-b-c\\}$ & $-a-b-c$ & $0$ & $a+2b+3c$ \\\\ \\hline \n\\end{tabular}\n\\end{center} \nObserve that $\\tau$ is dominated by both $\\cc_1^+$ and $\\cc_1^-$, and by definition $\\delta^+\\equiv 0$. Thus Theorem \\ref{trick} (iii) implies that $\\ind_{S^1}(\\e{(-\\tau)})=0$, as it\ncan also be checked directly from here \n\\begin{align*}\n\\ind_{S^1}(\\e{(-\\tau)})= &\\frac{1}{(1-t^{-a})(1-t^{-a-b})(1-t^{-a-b-c})}+\\frac{t^a}{(1-t^{a})(1-t^{-b})(1-t^{-b-c})}\\\\\n &+\\frac{t^{a+b}}{(1-t^{b})(1-t^{a+b})(1-t^{-c})}+\n\\frac{t^{a+b+c}}{(1-t^{c})(1-t^{b+c})(1-t^{a+b+c})}=0\\\\\n\\end{align*}\n\n \n\n\\end{exm}\n\n\n\\begin{rmk}\\label{index positive or negative}\nFollowing the discussion in Remark \\ref{ex 0 and c1}, by Theorem \\ref{trick} we have that if $\\tau\\in H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ satisfies $\\tau(p)\\geq 0$ (resp.\\;$\\tau(p)\\leq 0$)\nfor all $p\\in \\mathsf{M}^{S^1}$, then $\\ind_{S^1}(\\e{\\tau})\\in {\\mathbb{Z}}[t]$ (resp.\\;$\\ind_{S^1}(\\e{\\tau})\\in {\\mathbb{Z}}[t^{-1}]$).\n\\end{rmk}\n\nAs an immediate consequence of Theorem \\ref{trick}, we have the following\n\\begin{corollary}\\label{index 0 and -c1}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space with\n$\\mathsf{M}^{S^1}=\\{p_0,\\ldots,p_N\\}$. Let $N_i$ be the number of fixed points with exactly $i$ negative weights.\n\nIf $\\mathbf{1}\\in \\pic_{S^1}(\\mathsf{M})$ denotes the trivial line bundle over $\\mathsf{M}$, where $\\mathsf{c}_{1}^{S^1}(\\mathbf{1})=\\mathbf{0}$, then \n\\begin{equation}\\label{index 0}\n\\ind_{S^1}(\\mathbf{1})=N_0=N_n\\;.\n\\end{equation}\nIf $\\widetilde{\\mathbb{L}}^{S^1}\\in \\pic_{S^1}(\\mathsf{M})$ denotes the determinant line bundle $\\Lambda^n(T^*\\mathsf{M})$, where $\\mathsf{c}_{1}^{S^1}(\\widetilde{\\mathbb{L}}^{S^1})=\\mathsf{c}_{1}^{S^1}(\\Lambda^n(T^*\\mathsf{M}))=-\\mathsf{c}_{1}^{S^1}$, then\n\\begin{equation}\\label{index -c1}\n\\ind_{S^1}(\\widetilde{\\mathbb{L}}^{S^1})=(-1)^nN_0=(-1)^nN_n\\;.\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nAs we have already remarked, the classes $\\mathbf{0}$ and $\\mathsf{c}_{1}^{S^1}$ are dominated by $\\cc_1^+$ and $\\cc_1^-$. Thus \\eqref{index 0}\nand \\eqref{index -c1} follow from Theorem \\ref{trick} (iii) and the definition of $N_0$ and $N_n$. \n\\end{proof}\nNote that equation \\eqref{index 0} is already known, see for example \\cite[Corollary 2.7]{Ha} (also see \\cite[Theorem 2.3]{L}). \n\nObserve that \\eqref{index -c1} can also be obtained by noticing that since $\\mathsf{c}_{1}^{S^1}$ is dominated by $\\cc_1^+$ and $\\cc_1^-$, \n\\eqref{index integer} implies that \n$\\ind_{S^1}(\\widetilde{\\mathbb{L}}^{S^1})$ is an integer, thus $\\ind_{S^1}(\\widetilde{\\mathbb{L}}^{S^1})=\\ind(r_K(\\widetilde{\\mathbb{L}}^{S^1}))$, \nand so \\eqref{index -c1} follows from \\eqref{index symmetry} in Proposition \\ref{symmetries} and \\eqref{index 0}.\n\nWe also remark that $\\ind_{S^1}(\\mathbf{1})$ is the Todd genus of $\\mathsf{M}$; in fact\nfrom \\eqref{formula index 2} we have that \n\\begin{equation}\\label{todd genus}\n\\td(\\mathsf{M})= T_n[\\mathsf{M}]= \\ch(r_K(\\mathbf{1})) \\ttot[\\mathsf{M}]=\\ind(r_K(\\mathbf{1}))=\\ind_{S^1}(\\mathbf{1})\\,\n\\end{equation}\nwhere the second equality follows from observing that $\\ch(r_K(\\mathbf{1}))=1$, and the\nlast equality follows from \\eqref{K commutes} and the fact that $\\ind_{S^1}(\\mathbf{1})$ is an integer, thus $\\ind(r_K(\\mathbf{1}))=r_K(\\ind_{S^1}(\\mathbf{1}))=\\ind_{S^1}(\\mathbf{1})$.\nBy combining \\eqref{index 0} and \\eqref{todd genus} we recover the following well-known fact (see \\cite[Remark 2.10]{Ha} and \\cite{Fe}).\n\\begin{corollary}\\label{todd genus comp}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space,\n$N_i$ the number of fixed points with exactly $i$ negative weights, and $\\td(\\mathsf{M})$ the Todd genus of $\\mathsf{M}$. Then\n$$\n\\td(\\mathsf{M})=N_0=N_n.\n$$\n\\end{corollary}\nBefore giving the main application of Theorem \\ref{trick}, we prove the following easy but useful lemma. \n\\begin{lemma}\\label{c1 N0}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space, $\\mathsf{c}_1$ the first Chern class of the tangent bundle of $\\mathsf{M}$, $N_i$ the number of fixed points with exactly $i$ negative weights, and $\\td(\\mathsf{M})$ the Todd genus of $\\mathsf{M}$.\n\\begin{itemize}\n \\item[(a1)] If $\\eta\\in \\tor(H^2(\\mathsf{M},{\\mathbb{Z}}))$ then \n\\begin{equation}\\label{index torsion}\n\\ind(\\e{\\eta})=\\td(\\mathsf{M})=N_0\n\\end{equation}\nand\n \\begin{equation}\\label{torsion index}\n \\ind_{S^1}(\\e{\\eta^{S^1}})=t^{a}\\td(\\mathsf{M})=t^{a}N_0\\,,\n \\end{equation}\n where $\\eta^{S^1}\\in H_{S^1}^2(\\mathsf{M},{\\mathbb{Z}})$ denotes an equivariant extension of $\\eta$, and $a=\\eta^{S^1}(p)$ for every $p\\in \\mathsf{M}^{S^1}$.\\\\\n\\item[(a2)] If $\\mathsf{c}_1\\in \\tor(H^2(\\mathsf{M},{\\mathbb{Z}}))$ then $N_0=N_n=0$ and $\\td(\\mathsf{M})=0$.\n\\end{itemize}\n\n\\end{lemma}\n\\begin{proof}\n(a1) First of all, observe that if $\\eta\\in \\tor(H^2(\\mathsf{M},{\\mathbb{Z}}))$ then, by the discussion in Section \\ref{ecc}, it admits an equivariant extension $\\eta^{S^1}\\in H^2_{S^1}(\\mathsf{M},{\\mathbb{Z}})$. \nBy the commutativity of \\eqref{K commutes}, in order to prove \\eqref{index torsion} it is sufficient to prove \\eqref{torsion index}.\nIf $\\eta$ is torsion then there exists $k\\in {\\mathbb{Z}}\\setminus\\{0\\}$ such that $k\\eta=0$. Thus if we consider an equivariant extension \n$\\eta^{S^1}$, by \\eqref{trivial constant} we have that $\\eta^{S^1}(p)=a$ for some $a\\in {\\mathbb{Z}}$, for every $p\\in \\mathsf{M}^{S^1}$. Hence \n$$\n\\ind_{S^1}(\\e{\\eta^{S^1}})=t^{a}\\ind_{S^1}(\\mathbf{1})=t^{a}\\td(\\mathsf{M})=t^aN_0\n$$\nwhere the first equality follows from \\eqref{AS formula},\nthe second from \\eqref{todd genus}, and the last from Corollary \\ref{todd genus comp}.\n\n(a2)\nBy a similar argument, we have that the integer \n $\\mathsf{c}_{1}^{S^1}(p)$ does not depend on $p\\in M^{S^1}$. However $\\mathsf{c}_{1}^{S^1}(p_i)=\\sum_{j=1}^nw_{i,j}$, and by \\eqref{NiN} we have $N_0=N_n$.\nSo by definition of $N_0$ and $N_n$ we must have that\n$N_0=N_n=0$, and by Corollary \\ref{todd genus comp} that $\\td(\\mathsf{M})=0$.\n\n\\end{proof}\n\n\nThe next proposition also follows from Theorem \\ref{trick}, but it is a key result for the theorems in the next sections (see also \\cite[Assertion 4.10]{Ha} and \\cite[Proposition 2.5]{LL}).\n\\begin{prop}\\label{eq index zero}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space. Let $\\mathsf{c}_{1}^{S^1}$ be the equivariant first Chern class of the tangent bundle of $\\mathsf{M}$ and $k$ a positive integer such that \n$\\mathsf{c}_{1}^{S^1}(p)=k\\,\\eta^{S^1}(p)+c$ for all $p\\in \\mathsf{M}^{S^1}$, for some $\\eta^{S^1}\\in H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$ and $c\\in {\\mathbb{Z}}$.\nThen\n\\begin{equation}\\label{index 0 1..k0}\n \\ind_{S^1}(\\e{(-h\\eta^{S^1})})=0\\quad\\mbox{for every}\\quad h=1,\\ldots,k-1\\;.\n\\end{equation}\n\n\\end{prop}\n\\begin{rmk}\nObserve that if $\\mathsf{c}_1$ is torsion then $r_H(\\eta^{S^1})$ is also torsion, and by Lemma \\ref{c1 N0} it follows that \n \\begin{equation}\\label{index 0 always}\n \\ind_{S^1}(\\e{(-h\\eta^{S^1})})=0\\quad\\mbox{for every}\\quad h\\in {\\mathbb{Z}}\n \\end{equation}\n \\end{rmk}\n\\begin{proof}[Proof of Proposition \\ref{eq index zero}]\nFirst of all, observe that it is not restrictive to assume that $c=0$. In fact,\nlet $S^1\\times \\mathsf{M}\\to \\mathsf{M}$, $(\\lambda,q)\\to \\lambda\\cdot q$ be the given $S^1$-action on $\\mathsf{M}$, and consider \na new action given by $(\\lambda,q)\\to \\lambda^k\\cdot q$; we denote by $\\widetilde{S}^1$ the new circle acting on $\\mathsf{M}$.\nNote that the set of fixed points of this action coincides with the old one,\nand the new isotropy weights are the old ones multiplied by $k$. Thus \n $\\mathsf{c}_1^{\\widetilde{S}^1}(p)$ is divisible by $k$, for every $p\\in \\mathsf{M}^{\\widetilde{S}^1}=\\mathsf{M}^{S^1}$.\nSo there exists $\\widetilde{\\eta}\\in H_{\\widetilde{S}^1}^2(\\mathsf{M};{\\mathbb{Z}})$ such that\n$\\mathsf{c}_1^{\\widetilde{S}^1}=k\\widetilde{\\eta}$. Moreover, if $\\ind_{S^1}(\\e{\\eta^{S^1}})=P(t,t^{-1})$ for some $P\\in {\\mathbb{Z}}[x,y]$, then \n$\\ind_{\\widetilde{S}^1}(\\e{\\widetilde{\\eta}})=t^bP(t^k,t^{-k})$, for some $b\\in {\\mathbb{Z}}$. \nThus $\\ind_{S^1}(\\e{\\eta^{S^1}})=0$ if and only if $\\ind_{\\widetilde{S}^1}(\\e{\\widetilde{\\eta}})=0$.\nHence we can assume that $\\mathsf{c}_{1}^{S^1}(p)=k\\,\\eta^{S^1}(p)$ for all $p\\in \\mathsf{M}^{S^1}$.\n\nNotice that for all $p\\in \\mathsf{M}^{S^1}$ such that $\\eta^{S^1}(p)>0$ and all $h=1,\\ldots,k-1$, we have\n\\begin{equation}\\label{keta}\nh\\,\\eta^{S^1}(p)0$, and since $\\cc_1^+(p)$ is always nonnegative,\n$\\delta^+(p)=0$ for all $p\\in \\mathsf{M}^{S^1}$ such that $\\eta^{S^1}(p)\\neq 0$. \nFinally observe that if $\\eta^{S^1}(p)=\\cc_1^+(p)=0$, then $\\mathsf{c}_{1}^{S^1}(p)=0$ and $\\cc_1^-(p)=0$; however this is impossible, unless\n$\\dim(\\mathsf{M})=0$. So we can conclude that $\\delta^+(p)=0$ for all $p\\in \\mathsf{M}^{S^1}$.\n\nA similar argument shows that $h\\,\\eta^{S^1}$ is dominated by $\\cc_1^-$ for all $h=1,\\ldots,k-1$ (and $\\delta^-(p)=0$ for all $p\\in \\mathsf{M}^{S^1}$).\nSo the conclusion follows from Theorem \\ref{trick} (iii). \n\\end{proof}\n\n\\subsection{Symplectic manifolds} Suppose that $(\\mathsf{M},\\omega)$ is a compact, connected symplectic manifold endowed with a symplectic circle action with isolated fixed points. We recall that this triple is denoted by $(\\mathsf{M},\\omega,S^1)$. \nThe following lemma is a key fact to translate our results in the almost complex category to the symplectic category.\n\\begin{lemma}[\\cite{MD1}]\\label{N0 1}\nGiven $(\\mathsf{M},\\omega,S^1)$, \nthen $N_0$ can be either $0$ or $1$, and is $1$ exactly if the action is Hamiltonian.\n\\end{lemma}\n If the action is Hamiltonian, then $N_0$ coincides indeed with the number of points of minima of the moment map $\\psi$, which\n is $1$ because $\\psi$ is a Morse function with only even indices, and $\\mathsf{M}$ is assumed to be connected. \nMore in general, the equivariant perfection of $\\psi$ (see \\cite{Ki}) implies that\n\\begin{equation}\\label{bi=Ni}\nb_{2j}(\\mathsf{M})= N_j \\quad \\mbox{for every}\\quad j=0,\\ldots,n\\,, \n\\end{equation}\nwhere $b_{2j}(\\mathsf{M})$ denotes the $2j$-th Betti number of $\\mathsf{M}$. The following fact is a consequence of the results of this section:\n\\begin{lemma}\\label{Lemma:c1 not torsion}\nGiven $(\\mathsf{M},\\omega,S^1)$, if the action is Hamiltonian then $\\mathsf{c}_1$ is not a torsion\nelement in $H^2(\\mathsf{M};{\\mathbb{Z}})$. \n\\end{lemma}\n\\begin{proof}\nIt is sufficient to combine Lemma \\ref{N0 1} with Lemma \\ref{c1 N0} (a2). \n\\end{proof}\n\\begin{rmk}\\label{mcn}\nLet $(\\mathsf{M},\\omega)$ be a compact symplectic manifold with first Chern class $\\mathsf{c}_1$, and suppose it is not torsion.\nFollowing Definition 6.4.2 in \\cite{MDS}, the \\emph{minimal Chern number} of $(\\mathsf{M},\\omega)$ is defined to be the integer $N$ such that $\\langle \\mathsf{c}_1,\\pi_2(\\mathsf{M})\\rangle = N {\\mathbb{Z}}$.\nIf $\\mathsf{M}$ is simply connected then, by the Hurewicz theorem, we have $\\pi_2(\\mathsf{M})=H_2(\\mathsf{M},{\\mathbb{Z}})$ which, modulo torsion, is isomorphic to $H^2(\\mathsf{M},{\\mathbb{Z}})$, thus implying that the minimal Chern number agrees with the index of $(\\mathsf{M},\\omega)$. A result of Li \\cite{Li2} implies that \n if the $S^1$-action on $(\\mathsf{M},\\omega)$ is Hamiltonian with isolated fixed points then $\\mathsf{M}$ is simply connected. So it follows that\n if $(\\mathsf{M},\\omega)$ is endowed with a Hamiltonian $S^1$-action with isolated fixed points, the minimal Chern number always agrees with the index $\\k0$, which is not zero by Lemma \\ref{Lemma:c1 not torsion}. \\end{rmk}\n\n\\section{The Hilbert polynomial of $(\\mathsf{M},\\mathsf{J})$ and the equations in the Chern numbers}\\label{equations chern}\nWe recall from Section \\ref{ecc} that $\\mathcal{L}$ is the lattice given by $H^2(\\mathsf{M};{\\mathbb{Z}})\/\\tor(H^2(\\mathsf{M};{\\mathbb{Z}}))$ and $\\pi$ the projection $\\pi\\colon H^2(\\mathsf{M};{\\mathbb{Z}})\\to \\mathcal{L}$.\nIf $\\mathsf{c}_1$ is not torsion we have $\\pi(\\mathsf{c}_1)\\neq 0$, so there exists a non-torsion element $\\eta_0\\in H^2(\\mathsf{M};{\\mathbb{Z}})$ such that $\\pi(\\mathsf{c}_1)=\\k0\\,\\pi(\\eta_0)$.\nThe index $\\k0$, and when $\\k0>0$ the associated $\\eta_0\\in H^2(\\mathsf{M};{\\mathbb{Z}})$ (uniquely defined up to torsion), will play a crucial role in the rest of the section.\n\nBefore proceeding, we prove the following Lemma:\n\\begin{lemma}\\label{index torsion independent}\nLet $\\eta\\in H^2(\\mathsf{M};{\\mathbb{Z}})$ and $\\tau\\in \\tor(H^2(\\mathsf{M};{\\mathbb{Z}}))$. Then \n$$\n\\ind(\\e{(\\eta+\\tau)})=\\ind(\\e{\\eta})\\,.\n$$\n\\end{lemma}\n\\begin{proof}\nBy \\eqref{AT formula} we have that \n\n\\begin{align*}\n\\ind(\\e{(\\eta+\\tau)})& = \\ch(\\e{(\\eta+\\tau)})\\ttot[\\mathsf{M}]= \\left(1+\\eta+\\frac{\\eta^2}{2}+\\cdots\\right)\\left(1+\\tau+\\frac{\\tau^2}{2}+\\cdots\\right)\\ttot[\\mathsf{M}]=\\\\\n & = \\left(1+\\eta+\\frac{\\eta^2}{2}+\\cdots\\right)\\ttot[\\mathsf{M}]=\\ind(\\e \\eta)\\,,\n\\end{align*}\nwhere the second-last equality follows from the fact that if $\\tau$ is torsion then $ \\tau^k\\alpha[\\mathsf{M}]=0$ for all $k>0$ and $\\alpha\\in H^{2n-2k}(\\mathsf{M};{\\mathbb{Z}})$.\n\\end{proof}\n\nIn the rest of the section \\emph{we assume that $\\mathsf{c}_1$ is not torsion}. Let $\\eta_0\\in H^2(\\mathsf{M};{\\mathbb{Z}})$ be such that $\\pi(\\mathsf{c}_1)=\\k0 \\pi(\\eta_0)$. Even if $\\eta_0$\nis not uniquely defined, by Lemma \\ref{index torsion independent} the topological index $\\ind(\\e{\\eta})$ is independent on $\\eta\\in \\pi^{-1}(\\pi(\\eta_0))$. \nHence, given $(\\mathsf{M},\\mathsf{J})$ with $\\mathsf{c}_1$ not torsion, for every $k\\in {\\mathbb{Z}}$ the following integer\n\\begin{equation}\\label{polynomial}\n\\Hi(k)=\\ind(\\e{\\,k\\, \\eta_0})\n\\end{equation}\ndoes not depend on the choice of $\\eta_0$. Moreover,\nby \\eqref{AT formula} we obtain that\n\\begin{equation}\\label{HAT}\n\\Hi(k)= \\Big( \\sum_{h\\geq 0} \\frac{(k\\,\\eta_0)^h}{h!}\\Big)\\ttot[\\mathsf{M}]= \\sum_{h=0}^n k^h\\left( \\frac{\\mathsf{c}_1^h\\,T_{n-h}}{\\k0^h\\,h!}\\right)[\\mathsf{M}]\n\\end{equation}\nthus implying that, if $(\\mathsf{M},\\mathsf{J})$ has dimension $2n$, $\\Hi(k)$ is a polynomial in $k$ of degree at most $n$. \nThe polynomial $\\Hi(z)$ defined as\n\\begin{equation}\\label{Hilbert pol}\n\\Hi(z)= \\sum_{h=0}^n a_h z^h=\\sum_{h=0}^n \\left( \\frac{\\mathsf{c}_1^h\\,T_{n-h}}{\\k0^h\\,h!}[\\mathsf{M}]\\right)z^h, \\quad z\\in {\\mathbb{C}} \n\\end{equation}\nwill be referred to as\nthe \\emph{Hilbert polynomial of $(\\mathsf{M},\\mathsf{J})$}. \nThus \n\\begin{align}\n& a_n= \\frac{1}{\\k0^n\\,n!} \\mathsf{c}_1^n[\\mathsf{M}], \\;\\;\\;\\;\\;\\;a_{n-1}=\\frac{1}{2\\k0^{n-1}(n-1)!}\\mathsf{c}_1^n[\\mathsf{M}],\\nonumber \\\\\n\\label{ah} & a_{n-2}= \\frac{1}{12\\k0^{n-2}(n-2)!}(\\mathsf{c}_1^n+\\mathsf{c}_1^{n-2}\\mathsf{c}_2)[\\mathsf{M}]\\,, \\;\\;\\;\\;\n\\ldots\\\\\n& a_0= T_n[\\mathsf{M}] = \\td(\\mathsf{M}) \\nonumber\n\\end{align}\nThe first properties of $\\Hi(z)$ are given in the following \n\\begin{prop}\\label{properties P}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space with $N_0$ fixed points with zero negative weights.\nLet $\\mathsf{c}_1$ be the first Chern class of the tangent bundle of $\\mathsf{M}$ and assume that it is not torsion. Let $\\k0\\geq 1$ be the index of $(\\mathsf{M},\\mathsf{J})$,\n$\\Hi(z)$ the Hilbert polynomial, and $\\deg(\\Hi)$ its degree.\nThen \n\\begin{enumerate}\n \\item\\label{1a} $\\Hi(0)=\\td(\\mathsf{M})=N_0$;\\\\\n \\item\\label{3a} $\\Hi(z)=(-1)^n \\Hi(-\\k0-z)\\;\\;\\;$ for every $\\;\\;z\\in {\\mathbb{C}}$;\\\\\n \\item\\label{4a} $\\deg(\\Hi)\\equiv n \\mod 2$.\n\\end{enumerate}\n\n\\end{prop}\n\n\\begin{rmk}\\label{H Ham}\nBy Lemma \\ref{N0 1} and Proposition \\ref{properties P} \\eqref{1a}, note that if $(M,\\omega)$ is a compact symplectic manifold supporting\na Hamiltonian $S^1$-action with isolated fixed points, then the Hilbert polynomial $\\Hi(z)$ can never be identically zero.\n\\end{rmk}\n\n\\begin{rmk}\\label{properties Ch}\nProposition \\ref{properties P} \\eqref{4a} implies that if there exists $k$ such that $a_{n-2h}=0$ for every $h=0,\\ldots,k$, then\n$a_{n-2h-1}=0$ for every $h=0,\\ldots,k$.\n\\end{rmk}\n\n\\begin{proof}\nProperty \\eqref{1a} follows from the definition of $\\Hi(z)$ and Corollary \\ref{todd genus comp}. \nBy Lemma \\ref{line admissible}, every line bundle $\\mathbb{L}$ such that $\\pi(\\mathsf{c}_1(\\mathbb{L}))=k\\,\\pi(\\eta_0)$ is admissible. So from\nProposition \\ref{symmetries} we have that for all $k\\in {\\mathbb{Z}}$\n$$\n\\Hi(k)=\\ind(\\e{\\,k\\, \\eta_0})=(-1)^n \\ind(\\e{((-k-\\k0) \\eta_0)})=(-1)^n\\Hi(-\\k0-k)\\,,\n$$\nand \\eqref{3a} follows from observing that the polynomial given by $Q(z)=\\Hi(z)-(-1)^n \\Hi(-\\k0-z)$ is zero for all $k\\in {\\mathbb{Z}}$, hence it must be identically zero.\n\nIn order to prove \\eqref{4a} it is sufficient to notice that, if $\\Hi(z)=\\sum_{j=0}^ma_mz^m $, with $m=\\deg(\\Hi)$, from \\eqref{3a} it follows that\n$a_m=(-1)^{m+n}a_m$.\n\\end{proof}\nBefore proceeding with the main results of the section, we introduce some terminology that will be used in the discussion of the position of the roots of\n$\\Hi(z)$.\n\\begin{defin}\\label{def: RVGo}\nFix a positive integer $k$.\n\\begin{itemize}\n\\item[1)] We denote by $\\mathcal{T}_k$ the family of polynomials in $\\mathbb{R}[z]$ that can be written as $C(z)\\prod_{j=1}^{k-1}(z+j)$, where\n $C(z)\\in \\mathbb{R}[z]$ has all its roots on the line $l_{k}=\\{x+\\mathrm{i}y\\in {\\mathbb{C}}\\mid x=-\\frac{k}{2}\\}$. \n \\item[2)] We define $\\mathcal{S}_{k}$ to be the subset of the complex plane given by $$\\mathcal{S}_{k}= \\{x+\\mathrm{i}y\\in {\\mathbb{C}} \\mid -k 0}$. \nIn Sect.\\;\\ref{sec: generating fct} and Section \\ref{sec: values k0} we explore connections \namong our results and those in \\cite{RV}: we study under which conditions $\\Hi(z)$ belongs to $\\mathcal{T}_{\\k0}$, for certain values of $\\k0$.\nIn \\cite{Go}, Golyshev analyses the position of the roots of the Hilbert polynomial of a Fano variety and a variety of general type. In particular, after adapting\nhis terminology to ours, he asks under which conditions all\nthe zeros of $\\Hi(z)$ belong to the canonical strip $\\mathcal{S}_{\\k0}$.\nIn Section \\ref{sec: values k0} we will study the position of the roots of $\\Hi(z)$ in terms of inequalities in the Chern numbers and of $\\k0$, when $\\k0\\geq n-2$ (see Remarks \\ref{pos roots n+1}, \\ref{pos roots n} and Corollaries \\ref{pos roots n-1} and \\ref{pos roots n-2}).\n\nThe next corollary is a straightforward consequence of Proposition \\ref{properties P}. \n\\begin{corollary}\\label{property roots} \nLet $(\\M,\\J,S^1)$ be an $S^1$-space. Let $\\k0\\geq 1$ be the index of $(\\mathsf{M},\\mathsf{J})$, and\nassume that the Hilbert polynomial $\\Hi(z)$ is of positive degree $\\deg(\\Hi)>0$. If at least $\\deg(\\Hi)-3$ roots of $\\Hi(z)$, counted with\nmultiplicity, belong to $\\mathcal{C}_{\\k0}$, then all the roots of $\\Hi(z)$ belong to $\\mathcal{C}_{\\k0}$. In particular, if $n\\leq 3$, then all the roots of $\\Hi(z)$ belong to $\\mathcal{C}_{\\k0}$.\n\\end{corollary}\n\\begin{proof}\nLet $h$ be the number of roots, counted with multiplicity, which belong to $\\mathcal{C}_{\\k0}$; by assumption $h\\geq \\deg(\\Hi)-3$. \nSuppose that one of the remaining $\\deg(\\Hi)-h$ roots, $z_0\\in {\\mathbb{C}}$, does not belong to $\\mathcal{C}_{\\k0}$. Then, by Proposition \\ref{properties P} \\eqref{3a}, \nwe have that $z_1=-\\k0-z_0$ is also a root, and since $\\Hi(z)\\in \\mathbb{R}[z]$, the complex conjugates $z_2=\\overline{z_0}$ and $z_3=-\\k0-\\overline{z_0}$ are also roots.\nSince $z_0\\notin \\mathcal{C}_{\\k0}$, it follows that $z_i\\neq z_j$ for $i\\neq j$, and $z_i\\notin \\mathcal{C}_{\\k0}$ for $i=0,1,2,3$, implying that $\\Hi(z)$ has at least $h+4\\geq \\deg(\\Hi)+1$ roots, which is impossible\nsince we are assuming $\\Hi(z)$ to be non identically zero.\n\n\\end{proof}\n\n\nWe are now ready to prove Theorem \\ref{main theorem}.\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nChoose $\\eta_0$ and $\\tau$ in $H^2(\\mathsf{M};{\\mathbb{Z}})$ such that $\\mathsf{c}_1=\\k0 \\eta_0 + \\tau$,\nwhere $\\tau\\in \\tor(H^2(\\mathsf{M};{\\mathbb{Z}}))$. By Lemma \\ref{line admissible}, both $\\eta_0$ and $\\tau$ admit equivariant extensions\n$\\eta_0^{S^1}$ and $\\tau^{S^1}$ in $H_{S^1}^2(\\mathsf{M};{\\mathbb{Z}})$. Since $\\tau^{S^1}(p)$ does not depend on $p\\in \\mathsf{M}^{S^1}$ (see \\eqref{trivial constant}),\nit follows that $\\mathsf{c}_{1}^{S^1}(p)=\\k0 \\eta_0^{S^1}(p)+c$ for all $p\\in \\mathsf{M}^{S^1}$, for some $c\\in {\\mathbb{Z}}$.\nThus by Proposition \\ref{eq index zero} we have that\n\\begin{equation}\\label{key equation}\n\\ind_{S^1}(\\e{\\,k\\eta_0^{S^1}})=0 \\quad \\mbox{for all}\\quad k=-1,-2,\\ldots,-\\k0+1\\,,\n\\end{equation}\nand by combining \\eqref{K commutes} and \\eqref{key equation} we have that\n$$\n\\Hi(k)=\\ind(\\e{\\,k\\eta_0})=r_K(\\ind_{S^1}(\\e{\\,k\\eta_0^{S^1}}))=0 \\quad \\mbox{for all}\\quad k=-1,-2,\\ldots,-\\k0+1\\,,\n$$\nand \\eqref{H=0 even} follows.\n\nIn order to prove \\eqref{bound k0}, observe that by \\eqref{H=0 even} the set of roots of $\\Hi(z)$ contains $C_0=\\{-1,-2,\\ldots,-\\k0+1\\}$, thus if $\\Hi(z)\\not\\equiv 0$ we must have that $|C_0|=\\k0-1\\leq \\deg(\\Hi)\\leq n$.\n\\end{proof}\nNote that by Proposition \\ref{properties P}, $\\Hi(z)$\nhas a different behaviour depending on whether $N_0=0$ or not. \n\\begin{corollary}\\label{bound on k0}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space, and assume that $\\mathsf{c}_1$ is not torsion. Let $\\k0\\geq 1$ be the index of $(\\M,\\J,S^1)$ and\n$\\Hi(z)$ the Hilbert polynomial. Let\n$N_0$ be the number of fixed points with $0$ negative weights.\nThen:\n\\begin{itemize}\n \\item[({\\bf i})] If $N_0\\neq 0\\;\\;\\;$ then $\\;\\;\\;1\\leq \\k0\\leq \\deg(\\Hi)+1\\leq n+1$;\n \\item[({\\bf ii})] If $N_0=0\\;\\;\\;$ then either $\\;\\;\\deg(\\Hi)>0$ and $1\\leq \\k0\\leq \\deg(\\Hi)-1\\leq n-1$, or\n $\\Hi(z)\\equiv 0\n $, the latter being equivalent to $\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=0 \\quad \\mbox{for every}\\quad h=0,\\ldots,n\\,$.\n\\end{itemize}\n\\end{corollary}\n\\begin{proof}\nIf $\\k0=1$, the inequalities in ({\\bf i}) clearly hold. Assume $\\k0\\geq 2$.\nObserve that if $N_0\\neq 0$ then Proposition \\ref{properties P} \\eqref{1a} implies that $\\Hi(z)$ is not identically zero, and ({\\bf i}) follows from \\eqref{bound k0}. \n\nSuppose that $N_0=0$. Observe that in this case we must have\\footnote{Indeed, in Section \\ref{examples} it will be proved that $N_0=0$ implies $n\\geq 3$, see Prop.\\ \\ref{dim 4}.} $n\\geq 2$. Indeed, for $n=1$ it is impossible to have $N_0=0$, since by \\eqref{NiN} \nwe would have $N_0=N_1=0$, and hence $|\\mathsf{M}^{S^1}|=0$. \nBy Proposition \\ref{properties P} \\eqref{1a} and \\eqref{3a}, and by \\eqref{H=0 even}, we have that \nthe set of roots of \n$\\Hi(z)$ contains $C'_0=\\{0,-1,\\ldots,-\\k0\\}$. It follows that, if $\\Hi(z)$ is not identically zero, then $|C'_0|=\\k0+1\\leq \\deg(\\Hi)\\leq n$.\n\\end{proof}\nA consequence of Corollary \\ref{bound on k0} in the symplectic category is the following:\n\\begin{corollary}\\label{bound on k0 s}\nLet $(\\mathsf{M},\\omega)$ be a compact, connected symplectic manifold, and $\\k0$ the associated index. Then:\n\\begin{itemize}\n\\item[({\\bf i'})] If $(\\mathsf{M},\\omega)$ can be endowed with a Hamiltonian $S^1$-action with isolated fixed points, then $\\;\\;\\;1\\leq \\k0\\leq \\deg(\\Hi)+1\\leq n+1$;\n\\item[({\\bf ii'})] If $(\\mathsf{M},\\omega)$ can be endowed with a non-Hamiltonian $S^1$-action with isolated fixed points, then there are three possibilities:\n\\begin{itemize}\n\\item[(a)] $\\k0=0$, i.e.\\ $\\mathsf{c}_1$ is torsion;\n\\item[(b)] $\\k0>0$ and $\\Hi\\equiv 0$, the latter being equivalent to $\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=0 \\quad \\mbox{for every}\\quad h=0,\\ldots,n\\,$;\n\\item[(c)] $\\k0>0$, $\\deg(\\Hi)>0$ and $1\\leq \\k0\\leq \\deg(\\Hi)-1\\leq n-1$.\n\\end{itemize}\n\\end{itemize}\n\\end{corollary}\n\\begin{proof}\nIn order to prove ({\\bf i'}) it is sufficient to notice that, by Lemma \\ref{Lemma:c1 not torsion}, we must have $\\k0>0$. Then the claim follows from Lemma \\ref{N0 1} and Corollary \\ref{bound on k0} ({\\bf i}).\nThe only non trivial thing to prove in ({\\bf ii'}) is the upper bound on the index in (c). But this follows by combining Lemma \\ref{N0 1} and Corollary \\ref{bound on k0} ({\\bf ii}).\n\\end{proof}\nCorollary \\ref{minimal chern ham} follows from Corollary \\ref{bound on k0} ({\\bf i'}) and the discussion in Remark \\ref{mcn}.\n\\begin{rmk}\\label{rmk 1}\n(a') In the $6$-dimensional example $(\\widetilde{M},\\omega)$ constructed by Tolman \\cite{T3}, the image of $\\mathsf{c}_1^{S^1}(\\widetilde{\\mathsf{M}})$ under the restriction map $i^*\\colon H^2_{S^1}(\\widetilde{M};{\\mathbb{Z}})\\to H^2_{S^1}(\\widetilde{M}^{S^1};{\\mathbb{Z}})$\nis identically zero. Such restriction is zero when, for instance, $\\mathsf{c}_1$ is torsion in $H^2(\\mathsf{M};{\\mathbb{Z}})$ (see \\cite[Lemma 4.1]{GPS}). However, to the best of the author's knowledge,\nit is still not known whether\n$\\mathsf{c}_1(\\widetilde{M})$ is torsion. \n\\\\\n(b') Note that, under the hypothesis of ({\\bf ii'}), if $\\k0\\geq n$ then $\\Hi\\equiv 0$.\n\\end{rmk}\n\\begin{rmk}[{\\bf Comparison with Hattori's results}]\\label{hattori rmk}\nIn \\cite{Ha} Hattori analyses inequalities which are similar to those in Corollary \\ref{bound on k0}, provided that $(\\M,\\J,S^1)$ is an $S^1$-space endowed with a suitable quasi-ample line bundle, defined as follows.\nAn equivariant line bundle $\\mathbb{L}^{S^1}$ is \\emph{fine} if the restrictions of $\\mathbb{L}^{S^1}$ at the fixed points are mutually distinct\n$S^1$-modules, i.e.\\; if $\\mathbb{L}^{S^1}(p_i)= t^{a_i}\\neq t^{a_j}=\\mathbb{L}^{S^1}(p_j)$\nfor every $p_i\\neq p_j $ in $\\mathsf{M}^{S^1}$.\nIt is \\emph{quasi-ample} if it is fine and its first (non equivariant) Chern class satisfies $ \\mathsf{c}_1(\\mathbb{L}^{S^1})^n[\\mathsf{M}]\\neq 0$.\nIn \\cite[Theorem 5.1]{Ha} the author proves that if $(\\M,\\J,S^1)$ possesses a quasi ample line bundle $\\mathbb{L}^{S^1}$,\nand its first (non equivariant) Chern class satisfies $\\mathsf{c}_1=k\\, \\mathsf{c}_1(\\mathbb{L}^{S^1})$ for some\n$k\\in {\\mathbb{Z}}_{>0}$, then $k\\leq n+1 \\leq \\chi(\\mathsf{M})$. \nThus, if the equivariant line bundle $\\eta_0^{S^1}$ defined in the proof of Theorem \\ref{main theorem} is quasi-ample, Hattori's results\nimply that $\\k0\\leq n+1\\leq \\chi(\\mathsf{M})$. \nObserve that in Corollary \\ref{bound on k0} ({\\bf i}), we do not require the existence of a quasi-ample line bundle; we assume instead $N_0\\neq 0$.\nWe also remark that if $\\mathsf{c}_1^n[\\mathsf{M}]=0$ then $\\eta_0^{S^1}$ cannot be quasi-ample; on the other hand, if $ \\mathsf{c}_1^n[\\mathsf{M}]=0$ and $N_0\\neq 0$, Corollary \\ref{bound on k0} ({\\bf i}) gives a better upper bound on $\\k0$, since the vanishing of $ \\mathsf{c}_1^n[\\mathsf{M}]$ implies that $\\deg(\\Hi)\\leq n-2$, thus giving $\\k0\\leq n-1$ (see Remark \\ref{c1n=0}).\n\\end{rmk}\n\n\\begin{rmk}\\label{c1n=0}\nFrom \\eqref{ah}, Proposition \\ref{properties P} \\eqref{4a} and Corollary \\ref{bound on k0} it follows that if $\\k0\\geq 1$:\n\\begin{itemize}\n\\item If $ \\mathsf{c}_1^n[\\mathsf{M}]=0$ and $N_0\\neq 0$, then $\\deg(\\Hi(z))\\leq n-2$ and $ \\k0\\leq n-1$;\\\\\n\\item If $\\mathsf{c}_1^n[\\mathsf{M}]=0$ and $N_0=0$, then $\\k0\\leq n-3$ or $\\Hi(z)\\equiv 0$.\n\\end{itemize}\nSimilarly,\n\\begin{itemize}\n\\item If $\\mathsf{c}_1^n[\\mathsf{M}]=\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=0$ and $N_0\\neq 0$, then $\\deg(\\Hi(z))\\leq n-4$ and $ \\k0\\leq n-3$;\\\\\n\\item If $\\mathsf{c}_1^n[\\mathsf{M}]=\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=0$ and $N_0=0$, then $\\k0\\leq n-5$ or $\\Hi(z)\\equiv 0$.\n\\end{itemize}\n\\end{rmk}\n\n\\begin{rmk}\\label{nec non Ham}\nObserve that by Corollary \\ref{bound on k0 s} ({\\bf ii'}) and \\eqref{ah} it follows that if $(\\mathsf{M},\\omega)$ supports a non-Hamiltonian action and $\\k0\\geq n$, then\n$\\mathsf{c}_1^n[\\mathsf{M}]=\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=0$. In Theorem \\ref{nHam-char} we strengthen this fact and\n prove that for $(\\mathsf{M},\\omega,S^1)$ with $\\k0\\geq n$, the vanishing of one of these Chern numbers is indeed equivalent to having a non-Hamiltonian\naction. Moreover, if $\\k0=n-2$ or $\\k0=n-1$, then a suitable linear combination of those Chern number is zero if and only if the action is non-Hamiltonian. \n\\end{rmk} \nAs we have already observed before (Lemma \\ref{Lemma:c1 not torsion}), a compact symplectic manifold with $\\mathsf{c}_1$ torsion cannot support any Hamiltonian circle action.\nIf $\\mathsf{c}_1$ is not torsion, a criterion to conclude the same is given by the following\n\\begin{corollary}\\label{cor non ham 2}\nLet $(\\mathsf{M},\\omega)$ be a compact, connected symplectic manifold of dimension $2n$ with index $\\k0>0$.\nIf \n$$\n\\mathsf{c}_1^h\\,T_{n-h}[\\mathsf{M}]=0 \\quad \\mbox{for all}\\quad h\\geq 2\\k0-n+2\\Big\\lfloor\\frac{n-\\k0}{2}\\Big\\rfloor\n$$\nthen the manifold cannot support any Hamiltonian circle action with isolated fixed points. \n\\end{corollary}\n\\begin{proof}\nFirst of all observe that $\\;2\\k0-n+2\\Big\\lfloor\\frac{n-\\k0}{2}\\Big\\rfloor\\geq \\k0-1$, and equality holds if and only if $n\\not\\equiv \\k0\\mod{2}$.\nBy definition of Hilbert polynomial (see \\eqref{ah}),\nhaving\n$\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=0$ for all $h\\geq0$ implies that $\\Hi\\equiv 0$. If $\\k0\\geq 2$, having $\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=0$ for all $h\\geq \\k0-1$ implies\nthat $\\deg(\\Hi)\\leq \\k0-2$.\nHowever, as a consequence of Theorem \\ref{main theorem}, $\\Hi(z)$ has at least $\\k0-1$ zeroes, so $\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=0$ for all $h\\geq\\k0-1$ implies that\n$\\Hi\\equiv 0$. \n\nBy Remark \\ref{H Ham}, the Hilbert polynomial \nof a symplectic manifold with a Hamiltonian $S^1$-action and isolated fixed points can never be identically zero, and the corollary follows from the discussion above for $n\\not\\equiv \\k0\\mod{2}$.\nIf $n\\equiv \\k0\\mod{2}$ then, by Proposition \\ref{properties P} \\eqref{4a} we have that $\\deg(\\Hi)\\leq \\k0-1$ implies $\\deg(\\Hi)\\leq \\k0-2$, and the conclusion holds in this case too.\n\\end{proof}\n\nAnother consequence of Theorem \\ref{main theorem} is the following\n\\begin{corollary}\\label{extra root -k02}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space, and assume its index $\\k0$ is non-zero. Let $\\Hi(z)$ be the Hilbert polynomial. \n\\begin{equation}\\label{extra root}\n \\mbox{If} \\quad n\\equiv\\k0\\mod 2\\quad \\mbox{then} \\;\\;\\;\\Hi\\Big(-\\frac{\\k0}{2}\\Big)=0\\,.\n \\end{equation}\nMoreover, if $\\Hi(z)\\not\\equiv 0$ and $n\\equiv \\k0\\equiv 0\\mod 2$, then the multiplicity of the root $-\\frac{\\k0}{2}$ is at least $2$.\n\\end{corollary}\n\\begin{proof}\nObserve that, if $\\k0\\geq 2$, by \\eqref{H=0 even} we have that $\\widetilde{\\Hi}(z)=\\displaystyle\\frac{\\Hi(z)}{\\prod_{j=1}^{\\k0-1}(z+j)}$\nis a polynomial. The same conclusion follows if $\\k0=1$ by setting the empty product to be $1$. Hence\n by Proposition \\ref{properties P} \\eqref{3a} we have that for all $\\k0\\geq 1$\n\\begin{equation}\\label{H0 tilde}\n\\widetilde{\\Hi}(-\\k0-z)=\\frac{\\Hi(-\\k0-z)}{\\prod_{j=1}^{\\k0-1}(-\\k0-z+j)}=\\frac{(-1)^n\\Hi(z)}{(-1)^{\\k0-1}\\prod_{j=1}^{\\k0-1}(z+j)}=(-1)^{n-\\k0+1}\\widetilde{\\Hi}(z)\\;.\n\\end{equation}\nHence if $n\\equiv \\k0\\mod 2$, from \\eqref{H0 tilde} it follows that $\\widetilde{\\Hi}(-\\frac{\\k0}{2})=0$, thus proving \\eqref{extra root}.\nFinally, if $\\k0$ is even, then $-\\frac{\\k0}{2}\\in \\{-1,\\ldots,-\\k0+1\\}\\subset {\\mathbb{Z}}$, hence it is a root of both $\\prod_{j=1}^{\\k0-1}(z+j)$ and $\\widetilde{\\Hi}(z)$.\n\\end{proof}\n\nFrom Theorem \\ref{main theorem} we also have the following refinement of Corollary \\ref{property roots}, which concerns the position of the roots of $\\Hi(z)$.\n\\begin{corollary}\\label{position roots}\nLet $(\\M,\\J,S^1)$, $\\k0$ and $\\Hi(z)$ be as in Theorem \\ref{main theorem}, and assume that $\\deg(\\Hi)>0$. If $\\k0\\geq n-2$ then all the roots of $\\Hi(z)$ belong to $\\mathcal{C}_{\\k0}$. \n\\end{corollary}\n\nThe next corollary gives useful equations in the Chern numbers\ndepending on the index $\\k0$ and the parity of $n-\\k0$.\n\\begin{corollary}[{\\bf Equations in the Chern numbers}]\\label{cor equations chern numbers}\nLet $(\\M,\\J,S^1)$ be as in Theorem \\ref{main theorem}. Then \n\\begin{equation}\\label{equations chern numbers}\n\\sum_{h=0}^n \\frac{1}{h!}\\left( \\frac{k}{\\k0}\\right)^h \\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=0\\quad \\;\\;\\;\\;\\mbox{for all}\\;\\; k \\in \\{-1,-2,\\ldots,-\\k0+1\\}\\,.\n\\end{equation}\nMoreover, if $n\\equiv \\k0\\mod 2$ then\n\\begin{equation}\\label{k02}\n\\sum_{h=0}^n \\frac{(-1)^h}{2^h h!}\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=0\\;,\n\\end{equation}\nand if $n\\equiv \\k0 \\equiv 0\\mod 2$ then\n\\begin{equation}\\label{k02 2}\n\\sum_{h=1}^n \\frac{(-1)^{h-1}}{2^{h-1} (h-1)!}\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=0\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nIt is sufficient to notice that \\eqref{k02 2} is equivalent to having $\\Hi'(-\\frac{\\k0}{2})=0$, and the proof of Corollary \\ref{cor equations chern numbers} is a direct consequence of Theorem \\ref{main theorem}, Corollary \\ref{extra root -k02} and\nthe definition of Hilbert polynomial\n\\eqref{Hilbert pol}.\n\\end{proof}\nThus the cases in which we can derive more restrictions on the Chern numbers are when $\\k0$ is ``large'' (see Section \\ref{sec: values k0}). \n\\\\$\\;$\n\nBefore proceeding with the analysis of $\\Hi(z)$ for different values of $\\k0$, in the next subsection we study the properties of\nthe generating function of the sequence $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$.\n\n\n\\subsection{The generating function associated to the Hilbert polynomial}\\label{sec: generating fct}\n\nWe recall that the \\emph{generating function} of a sequence $\\{b_k\\}_{k\\in \\mathbb{N}}\\subset \\mathbb{R}$ is the formal power series\n$$\nP(t)=\\sum_{k\\geq 0} b_k t^k\\,.\n$$\n\n\nThe following result is due to Popoviciu \\cite{Po} (see also \n\\cite[Corollary 4.7]{St}).\n\\begin{prop}[Popoviciu]\\label{Pop}\nLet $H(z)$ be a polynomial of degree $m$ and $P(t)$ the generating function of the sequence $\\{H(k)\\}_{k\\in \\mathbb{N}}$. Then \n\\begin{equation}\\label{sym P}\nP(t^{-1})=(-1)^{m+1}t^{k_0}P(t) \n \\end{equation}\n for some $k_0\\in {\\mathbb{Z}}$ if and only if $k_0\\geq 1$, \n \\begin{equation}\\label{zeros H}\n H(-1)=H(-2)=\\cdots = H(-k_0+1)=0\n \\end{equation}\n and \n \\begin{equation}\\label{symmetry}\n H(k)=(-1)^m H(-k_0-k)\\quad \\mbox{for every}\\quad k\\in {\\mathbb{Z}}\\,.\n \\end{equation}\n\\end{prop}\n\nAs a consequence of the properties satisfied by $\\Hi(z)$, we have the following\n\n\\begin{prop}\\label{gen fct hilbert}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space and assume its index is non-zero. Let $\\Hi(z)$ be the associated Hilbert polynomial of degree $\\deg(\\Hi)=m$. Let $N_0$\nbe the number of fixed points with $0$ negative weights.\nThen the generating function $\\Gen(t)$ of $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$ \nis given by\n\\begin{equation}\\label{gener fct}\n\\Gen(t)=\\frac{\\U(t)}{(1-t)^{m+1}} \n\\end{equation}\nwhere $\\U(t)$ is a polynomial in $\\mathbb{R}[t]$ such that $\\U(0)=N_0$, with \n\\begin{equation}\\label{prop Gen1}\n\\Gen(t^{-1})=(-1)^{m+1}t^{\\k0} \\Gen(t)\n\\end{equation}\nand \n\\begin{equation}\\label{prop U}\n \\U(t^{-1})=t^{\\k0-m-1}\\U(t)\\,.\n\\end{equation}\nMoreover, if $\\Hi(z)\\not\\equiv 0$, then\n\\begin{equation}\\label{degree U}\n\\frac{m+1-\\k0}{2}\\leq \\deg(\\U)\\leq m+1-\\k0\\,,\n\\end{equation}\nand $\\deg(\\U)=m+1-\\k0$ if and only if $N_0\\neq 0$.\nHere $\\deg(\\U)$ denotes the degree of $\\U$.\n\\end{prop}\nThus, by Lemma \\ref{N0 1}, if $(\\mathsf{M},\\omega)$ is a compact symplectic manifold and the $S^1$-action is Hamiltonian, then \nthe polynomial $\\U(t)$ is of degree $m+1-\\k0$.\n\\begin{proof}\nIt is well known that the generating function of a sequence $\\{H(k)\\}_{k\\in \\mathbb{N}}$, where $H\\in \\mathbb{R}[z]$ is a polynomial of degree $m$,\nis of the form given by \\eqref{gener fct}, where $\\U(t)\\in \\mathbb{R}[t]$ is a polynomial of degree at most equal to $m$. In order to prove that $\\U(0)=N_0$, observe that \n$$\\Gen(t)=\\frac{\\U(t)}{(1-t)^{m+1}}= \\U(t)\\sum_{k\\geq 0}\\binom{m+k}{m}t^k=\\U(0)+tQ(t)$$\nfor some formal power series $Q(t)\\in \\mathbb{R}[[t]]$. Thus $\\U(0)=\\Gen(0)=\\Hi(0)$, and by Proposition \\ref{properties P} \\eqref{1a} $\\Hi(0)=N_0$.\n\nAs for \\eqref{prop U}, observe that by Theorem \\ref{main theorem} \\eqref{H=0 even}, if $\\k0\\geq 2$ we have that \\eqref{zeros H} is satisfied for $k_0=\\k0$, the index of $(\\mathsf{M},\\mathsf{J})$.\nIf $\\k0=1$ \\eqref{zeros H} is trivially satisfied, since it is the empty condition. \nMoreover, by Proposition \\ref{properties P} \\eqref{3a} and \\eqref{4a}, we have that \\eqref{symmetry} is satisfied as well. Thus by Proposition \\ref{Pop} \nthe generating function $\\Gen(t)$ of $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$ satisfies \\eqref{prop Gen1}, obtaining\n$$\n(-1)^{m+1}\\frac{t^{m+1}\\U(t^{-1})}{(1-t)^{m+1}}=\\Gen(t^{-1})=(-1)^{m+1}t^{\\k0}\\Gen(t)=(-1)^{m+1}\\frac{t^{\\k0}\\U(t)}{(1-t)^{m+1}}\n$$\nand \\eqref{prop U} follows.\n\nLet $e=\\deg(\\U)$ and $\\U(t)=\\alpha_0+\\alpha_1t+\\cdots +\\alpha_e t^e$. By \\eqref{prop U} we have that\n\\begin{equation}\\label{coeff U}\n\\alpha_et^{m+1-\\k0-e}+\\alpha_{e-1}t^{m+1-\\k0-e+1}+\\cdots + \\alpha_0t^{m+1-\\k0}=\\alpha_0+\\alpha_1t+\\cdots +\\alpha_e t^e\\,,\n\\end{equation}\nhence we must have $0\\leq m+1-\\k0-e\\leq e$, and \\eqref{degree U} follows.\nThe equality in \\eqref{coeff U} also implies that $\\alpha_0=\\U(0)=N_0\\neq 0$ if and only if $m+1-\\k0-e=0$.\n\\end{proof}\n\nWe recall that a polynomial of degree $e$, $U(t)=\\alpha_0+\\alpha_1t+\\cdots + \\alpha_e t^e$, is called \\emph{self-reciprocal} if \n\\begin{equation}\\label{self-rec}\nt^{e}U(t^{-1})=U(t)\\,.\n\\end{equation}\nSuch a polynomial is sometimes also referred to as a \\emph{palindromic}, since \\eqref{self-rec} is equivalent to saying that\nthe list of coefficients $\\alpha_0\\,\\alpha_1\\,\\cdots \\alpha_e$ is a palindrome, i.e.\\;$\\alpha_i=\\alpha_{e-i}$ for every $i$.\n\\begin{corollary}\\label{U palindrom}\nWith the same notation of Proposition \\ref{gen fct hilbert}, we have that:\n\\begin{itemize}\n\\item[({\\bf i})] $\\U(t)$ is divisible by $t^{m+1-\\k0-e}$, where\n$e=\\deg(\\U)$, and the polynomial $t^{e+\\k0-m-1}\\U(t)$ is self-reciprocal. \n\\item[({\\bf ii})] If $(\\mathsf{M},\\omega)$ is a symplectic manifold and the $S^1$-action is Hamiltonian,\nthen $\\U(t)$ is self-reciprocal. Moreover if $(\\mathsf{M},\\omega)$ is monotone with $\\mathsf{c}_1=\\k0 [\\omega]$, then \n$\\deg(\\U)=n+1-\\k0$.\n\\end{itemize}\n\\end{corollary}\n\\begin{proof}\nThe claims in ({\\bf i}) are a consequence of \\eqref{prop U} and \\eqref{coeff U}. \nIf $\\deg(\\U)=e=m+1-\\k0$, which by Proposition \\ref{gen fct hilbert} is equivalent to having $N_0\\neq 0$,\nwe obtain that $\\U(t)$ is self-reciprocal, and the first claim in ({\\bf ii}) follows from Lemma \\ref{N0 1}.\nThe second claim follows from observing that monotonicity implies $ \\mathsf{c}_1^n[\\mathsf{M}] \\neq 0$, hence $\\deg(\\Hi)=n$.\n\\end{proof}\n\\begin{rmk}\\label{num of cds}\nObserve that the polynomial $\\U(t)$ determines $\\Gen(t)$ which, in turns, determines $\\Hi(z)$. Thus the Hilbert polynomial, and hence\nall the combinations of Chern numbers $\\mathsf{c}_1^h T_{n-h}[\\mathsf{M}]$, for $h=0,\\ldots,n$, are completely determined by the coefficients of $\\U(t)$.\nMoreover, if $N_0$ is given, the coefficient of degree zero in $\\U(t)$ is known, since by Proposition \\ref{gen fct hilbert} $\\U(0)=N_0$.\nIn conclusion, from Corollary \\ref{U palindrom} it follows that the number of coefficients of $\\U(t)$ to determine is at most\nequal to $\\floor*{\\displaystyle\\frac{m-\\k0-1}{2}}+1$. This explains why\nthe number of conditions that completely determine the Hilbert polynomial (and hence the combinations of Chern numbers $\\mathsf{c}_1^h T_{n-h}[\\mathsf{M}]$)\nis the same when $\\k0=n+1-2k$ and $\\k0=n-2k$, for every $k\\in {\\mathbb{Z}}$ such that $0\\leq k\\leq \\frac{n-1}{2}$.\n\\end{rmk}\nIn the beautiful note \\cite{RV}, the author analysis the position of the roots of $\\Hi(z)$ in terms of those of $\\U(t)$,\n deriving the following\n\\begin{thm}[Rodriguez-Villegas \\cite{RV}]\\label{RV theorem}\nLet the notation be as in Proposition \\ref{gen fct hilbert}, and $\\mathcal{T}_k$ as in Definition \\ref{def: RVGo}. \nAssume that $\\Hi(z)\\not\\equiv 0$ and that all the roots of $\\U(t)$ are on the unit circle. Then $\\Hi(z)$ belongs to $\\mathcal{T}_{\\k0}$.\n\\end{thm}\nIn the next section we analyse the different expressions of $\\U(t)$ for $\\k0\\in \\{n-2,n-1,n,n+1\\}$.\nAs a consequence, we prove that if $\\k0=n$ or $\\k0=n+1$, then $\\Hi(z)$ always belongs to $\\mathcal{T}_{\\k0}$ (unless $\\Hi(z)\\equiv 0$).\nIf $\\k0=n-2$ or $n-1$, we\nderive necessary and sufficient conditions on the Chern numbers that \nensure $\\Hi(z)$ to be in $\\mathcal{T}_{\\k0}$, or more in general that ensure its roots to be on the canonical strip $\\mathcal{S}_{\\k0}$ (see Corollaries \\ref{pos roots n-1} and \\ref{pos roots n-2}). As a byproduct, we prove that when $N_0=1$ and $n$ is big enough, then $\\Hi(z)$ belongs to $\\mathcal{T}_{\\k0}$ \\emph{if and only if}\nthe roots of $\\U(t)$ are on the unit circle (see Corollaries \\ref{RV1} and \\ref{RV2}).\n\n\\subsection{Connection with Ehrhart polynomials}\\label{connections ehrhart}\nSome of the results in Section \\ref{equations chern} can be regarded as a generalisation of what is already known for the Ehrhart polynomial of a reflexive polytope.\nThe link between Hilbert polynomials of $S^1$-spaces and Ehrhart polynomials of reflexive polytopes is given by monotone symplectic toric manifolds.\n\nSuppose that $(\\mathsf{M},\\omega)$ is a compact symplectic manifold of dimension $2n$, and that the $S^1$-action extends to a toric action, i.e.\\\n$S^1$ is a circle subgroup in an $n$-dimensional torus $\\mathbb{T}^n$ which is acting effectively on $(\\mathsf{M},\\omega)$ \n with moment map $\\Psi\\colon (\\mathsf{M},\\omega) \\to Lie(\\mathbb{T}^n)^*$. We identify $Lie(\\mathbb{T}^n)^*$ with $\\mathbb{R}^n$, and let the dual lattice of $\\mathbb{T}^n$ be ${\\mathbb{Z}}^n$. \n By the Atiyah \\cite{At1} and Guillemin-Sternberg \\cite{GS82} convexity theorem, we know\n that $\\Psi(\\mathsf{M})=: \\Delta$ is a convex polytope, more precisely it is the convex hull \n of its vertices, which coincide with the images of the fixed points of the $\\mathbb{T}^n$ action. \n Suppose that $(\\mathsf{M},\\omega)$ is also \\emph{monotone} and rescale the symplectic form so that $\\mathsf{c}_1=\\k0 [\\omega]$ (so $[\\omega]$\n is primitive in $H^2(\\mathsf{M};{\\mathbb{Z}})$, which is torsion free in this case). Choose the moment map $\\Psi$ so that all the vertices of $\\Delta$ belong to the lattice ${\\mathbb{Z}}^n$: we call such polytope $\\Delta$ \\emph{primitive} and \\emph{integral}. \n As a consequence of a result of Danilov \\cite{Da}, we have that \n \\emph{the Hilbert polynomial $\\Hi(z)$ of $(\\mathsf{M},\\omega)$ coincides with the Ehrhart polynomial $i_{\\Delta}(z)$ of $\\Delta$}.\n Moreover, it is well-known that there exists a (unique) $k\\in {\\mathbb{Z}}_{>0}$ such that the dilated polytope \n $\\Delta'=k \\Delta$, suitably translated by an integer vector, is \\emph{reflexive}\\footnote{An integral polytope $\\mathcal{P}\\subset \\mathbb{R}^n$ of dimension $n$ is reflexive if it contains the origin in its interior, and its dual polytope $\\mathcal{P}^*=\\{\\mathbf{x}\\in \\mathbb{R}^n\\mid \\mathbf{x}\\cdot \\mathbf{y}\\geq -1\\mbox{ for all }\\mathbf{y}\\in \\mathcal{P}\\}$\n is also integral.}. By a result of Hibi \\cite{Hibi}, this is equivalent to saying that the Ehrhart polynomial $i_{\\Delta'}(z)$ and its associated generating function $P_\\Delta'(t)=\\frac{U(t)}{(1-t)^{n+1}}$\n satisfy \n \\begin{equation}\\label{deltas}\n i_{\\Delta'}(z)=(-1)^n i_{\\Delta'}(-1-z)\\quad\\quad\\mbox{and}\\quad\\quad P_{\\Delta'}(t^{-1})=(-1)^{n+1}t \\,P_{\\Delta'}(t).\n \\end{equation}\nThe following gives a combinatorial characterisation of the index $\\k0$ of $(\\mathsf{M},\\omega)$ (which, by Remark \\ref{mcn}, coincides with the minimal Chern number): \n\\begin{lemma}\\label{k0 equivalence}\nLet $(\\mathsf{M},\\omega,\\mathbb{T},\\Psi)$ be a monotone symplectic toric manifold, with symplectic form satisfying $\\mathsf{c}_1=\\k0[\\omega]$. Consider the primitive integral moment polytope image $\\Delta$.\nThen the index $\\k0$ is the unique integer so that $\\Delta'=\\k0 \\Delta$ is reflexive.\n\\end{lemma}\n\\begin{proof}\nFirst of all observe that, from $\\Delta'=k\\Delta$ we have $i_{\\Delta}(z)=i_{\\Delta'}\\big(\\frac{z}{k}\\big)$ for every $z\\in {\\mathbb{C}}$.\nMoreover, as mentioned before, $\\Hi(z)=i_\\Delta(z)$.\n So from \\eqref{deltas} we have that \n$$\n\\Hi(z)=i_{\\Delta}(z)=i_{\\Delta'}\\Big(\\frac{z}{k}\\Big)=(-1)^n i_{\\Delta'}\\Big(-1-\\frac{z}{k}\\Big)=(-1)^n i_\\Delta(-k-z)=(-1)^n \\Hi(-k-z)\\,,\n$$\nfor every $z\\in {\\mathbb{C}}$. By Remark \\ref{H Ham}, $\\Hi(z)$ is a nonzero polynomial, so Proposition \\ref{properties P} \\eqref{3a} implies that $\\k0=k$. \n\\end{proof}\nIt is in this sense that we can regard the symmetry property of $\\Hi(z)$ (i.e.\\ Proposition \\ref{properties P} \\eqref{3a}) and the results in Proposition \\ref{gen fct hilbert} as a generalisation of \n\\eqref{deltas}.\n\n\\section{Computation of $\\Hi(z)$ and Chern numbers for some values of $\\k0$}\\label{sec: values k0}\nIn this section, we compute explicitly the Hilbert polynomial $\\Hi(z)$ and its associated generating \nfunction for $\\k0\\geq n-2$ and $\\k0\\neq 0$, deriving more properties of the Chern numbers of $(\\M,\\J,S^1)$. \n\nLet $\\sigma_j(x_1,\\ldots,x_n)$ be the $j$-th elementary symmetric polynomials \nin $x_1\\ldots,x_n$, for $j=0,\\ldots,n$, and let $\\left[ \\begin{array}{c} n \\\\ k \\end{array} \\right]$ be the \\emph{unsigned Stirling numbers of the\nfirst kind}, where $k,n \\in \\mathbb{N}$ and $1\\leq k\\leq n$, satisfying\n\\begin{equation}\\label{stirling}\n(x)^{(n)}=x(x+1)\\cdots (x+n-1)=\\sum_{k=0}^n \\left[ \\begin{array}{c} n \\\\ k \\end{array} \\right]x^k\\,,\n\\end{equation}\nwhere $(x)^{(n)}$ is the rising factorial. \nThus we have the relation:\n\\begin{equation}\\label{stirling permutation}\n\\sigma_k(1,2,\\ldots,n)=\\left[ \\begin{array}{c} n+1 \\\\ \\\\ n-k+1 \\end{array} \\right]\\,,\n\\end{equation}\nand the following well-known identities:\n\\begin{align}\n& \\sigma_0(1,2,\\ldots,n)=\\left[ \\begin{array}{c} n+1 \\\\ n+1 \\end{array} \\right]=1\\nonumber\\\\\n\\label{sigma1}& \\sigma_1(1,2,\\ldots,n)=\\left[ \\begin{array}{c} n+1 \\\\ n \\end{array} \\right] = \\binom{n+1}{2}\\\\\n\\label{sigma2} & \\sigma_2(1,2,\\ldots,n)= \\left[ \\begin{array}{c} n+1 \\\\ n-1 \\end{array} \\right] =\\frac{1}{4}(3n+2)\\binom{n+1}{3}=\\frac{(3n+2)(n+1)n(n-1)}{24}\n\\end{align}\nObserve that by Corollary \\ref{bound on k0}, if $\\k0>n+1$ then $\\Hi(z)\\equiv 0$ and $ \\mathsf{c}_1^h T_{n-h}[\\mathsf{M}]=0$ for every $h=0,\\ldots,n$. So in the rest of the section\nwe will focus on the cases in which $0<\\k0\\leq n+1$.\n\nBefore beginning, we remind the reader that the Hilbert polynomial of ${\\mathbb{C}} P^n$ is given by $\\frac{\\prod_{j=1}^n(z+j)}{n!}$.\n\\begin{prop}[$\\k0=\\mathbf{n+1}$]\\label{cor n+1}\n\nLet $(\\M,\\J,S^1)$ be and $S^1$-space with index $\\k0=n+1$. Let $N_0$ be the number of fixed points with $0$ negative weights.\n Then \n\\begin{equation}\\label{H k0=n+1}\n\\Hi(z)= \\frac{N_0}{n!}\\prod_{j=1}^n(z+j)=N_0 \\Hi_{{\\mathbb{C}} P^n}(z)\\,,\n\\end{equation}\nwhere $\\Hi_{{\\mathbb{C}} P^n}(z)$ is the Hilbert polynomial of ${\\mathbb{C}} P^n$, and for every $h=0,\\ldots,n$ we have \n\\begin{equation}\\label{n+1 precise}\n\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=N_0\\frac{h!(n+1)^h}{n!}\\left[ \\begin{array}{c} n+1 \\\\ h+1 \\end{array} \\right]=N_0\\, \\mathsf{c}_1^h\\,T_{n-h}[{\\mathbb{C}} P^n].\n\\end{equation}\nIn particular \n\\begin{equation}\\label{c1 n+1}\n\\mathsf{c}_1^n[\\mathsf{M}]=N_0 (n+1)^n\\, \n\\end{equation}\nand\n\\begin{equation}\\label{c1c2}\n \\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=N_0\\frac{n(n+1)^{n-1}}{2}\\,.\n\\end{equation}\nMoreover, the generating function of $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$ is given by\n\\begin{equation}\\label{gen fct n+1}\n\\Gen(t)=N_0\\frac{1}{(1-t)^{n+1}}\n\\end{equation}\n\\end{prop}\n\n\\begin{rmk}\\label{pos roots n+1}\nFrom \\eqref{gen fct n+1} and Proposition \\ref{gen fct hilbert} we have that in this case $\\U(t)=N_0$, and\nif $N_0\\neq 0$, the zeros of $\\Hi(z)$ coincide with the integers greater than $-\\k0=-(n+1)$ and smaller than $0$,\nthus in particular $\\Hi(z)$ belongs to $\\mathcal{T}_{n+1}$, and hence all its roots are on the canonical strip $\\mathcal{S}_{n+1}$ (see Theorem \\ref{RV theorem}).\n\\end{rmk}\n\n\n\\begin{proof}[Proof of Proposition \\ref{cor n+1}]\nIf $N_0=0$ then all the claims in Proposition \\ref{cor n+1} follow from Corollary \\ref{bound on k0} ({\\bf ii}). \nSuppose that $N_0\\neq 0$. By Proposition \\ref{properties P} \\eqref{1a}, $\\Hi(z)$ is a nonzero polynomial which, by Theorem \\ref{main theorem}\n\\eqref{H=0 even}, has roots $-1,-2,\\ldots,-n$ (note that in this case $\\k0\\geq 2$). Thus $\\Hi(z)=\\alpha \\prod_{j=1}^n(z+j)$.\nIn order to find $\\alpha$ we can use Proposition \\ref{properties P} \\eqref{1a}, obtaining $\\Hi(0)=\\alpha\\, n!=N_0$, and \\eqref{H k0=n+1} follows.\nFor $h=0,\\ldots,n$, the term of degree $h$ on the right hand side of \\eqref{H k0=n+1} is given by $\\frac{N_0}{n!}\\sigma_{n-h}(1,2,\\ldots,n)=\\frac{N_0}{n!}\\left[ \\begin{array}{c} n+1 \\\\ h+1 \\end{array} \\right]$.\nOn the other hand, the term of degree $h$ on the left hand side of \\eqref{H k0=n+1} can by computed by using \\eqref{Hilbert pol}, obtaining\n$\\displaystyle\\frac{\\mathsf{c}_1^h\\,T_{n-h}}{(n+1)^h\\,h!}[\\mathsf{M}]$; this completes the proof of \\eqref{n+1 precise}.\nIn order to prove \\eqref{c1 n+1} it is sufficient to consider \\eqref{n+1 precise} with $h=n$ (or $h=n-1$).\nBy taking $h=n-2$, from \\eqref{n+1 precise} we have \n\\begin{equation}\\label{sigma 2}\n\\mathsf{c}_1^{n-2}\\left(\\frac{\\mathsf{c}_1^2+\\mathsf{c}_2}{12}\\right)[\\mathsf{M}]=N_0 \\frac{(n-2)!(n+1)^{n-2}}{n!}\\left[ \\begin{array}{c} n+1 \\\\ n-1 \\end{array} \\right]\\,,\n\\end{equation} \nwhich, combined with \\eqref{c1 n+1} and \\eqref{sigma2} proves \\eqref{c1c2}.\nIn order to prove \\eqref{gen fct n+1}, observe that, by the above discussion, if $\\k0=n+1$ then $\\Hi(z)$ is either of degree $n$,\nwhich happens exactly if $N_0\\neq 0$, or it is identically zero. In the first case, by Proposition \\ref{gen fct hilbert}, $\\U(t)$ is of degree\nzero and $\\U(0)=N_0$, implying \\eqref{gen fct n+1}.\n\\end{proof}\n\n\nAs we will see in the next proposition, the case $\\k0=n$ is similar to $\\k0=n+1$.\nWe recall that the Hilbert polynomial of $Q$, the hyperquadric in ${\\mathbb{C}} P^{n+1}$, is given by $\\frac{2}{n!}\\Big(z+\\frac{n}{2}\\Big)\\prod_{j=1}^{n-1}(z+j)$.\n\\begin{prop}[$\\k0=\\mathbf{n}$]\\label{cor n}\n\nLet $(\\M,\\J,S^1)$ be and $S^1$-space with index $\\k0=n$. Let $N_0$ be the number of fixed points with $0$ negative weights.\nThen $n\\geq 2$ and\n\\begin{equation}\\label{H k0=n}\n\\Hi(z)= \\frac{2\\,N_0}{n!}\\Big(z+\\frac{n}{2}\\Big)\\prod_{j=1}^{n-1}(z+j)=N_0 \\Hi_Q(z)\\,, \n\\end{equation}\nwhere $\\Hi_Q(z)$ is the Hilbert polynomial of $Q$, the hyperquadric in ${\\mathbb{C}} P^{n+1}$. \nThus for every $h=0,\\ldots,n$ we have \n\\begin{equation}\\label{n precise}\n\\mathsf{c}_1^h\\, T_{n-h}[\\mathsf{M}]=N_0\\frac{2\\,h!\\,n^h}{n!}\\Big(\\left[ \\begin{array}{c} n \\\\ h \\end{array} \\right]+\\frac{n}{2}\\left[ \\begin{array}{c} n \\\\ h+1 \\end{array} \\right]\\Big)\\, = N_0 \\, \\mathsf{c}_1^h\\,T_{n-h}[Q].\n\\end{equation}\nIn particular \n\\begin{equation}\\label{c1 n}\n\\mathsf{c}_1^n[\\mathsf{M}]=N_0\\, 2 n^n\n\\end{equation}\nand\n\\begin{equation}\\label{c1c22}\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=N_0\\,n^{n-2}(n^2-n+2)\\,.\n\\end{equation}\nMoreover, the generating function of $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$ is given by\n\\begin{equation}\\label{gen fct n}\n\\Gen(t)=N_0\\frac{1+t}{(1-t)^{n+1}}\n\\end{equation}\n\\end{prop}\n\n\n\\begin{rmk}\\label{pos roots n}\nFrom \\eqref{gen fct n} and Proposition \\ref{gen fct hilbert} we have that in this case $\\U(t)=N_0(1+t)$. Thus,\nif $N_0\\neq 0$, the root of $\\U(t)$ is on the unit circle, and \nthe zeros of $\\Hi(z)$ coincide with the integers greater than $-\\k0=-n$ and smaller than $0$, together with $-\\frac{n}{2}$,\nthus in particular $\\Hi(z)$ belongs to $\\mathcal{T}_{n}$, and hence its roots are on the canonical strip $\\mathcal{S}_{n}$ (see Theorem \\ref{RV theorem}).\n\n\\end{rmk}\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{cor n}]\nFirst of all, for $n=1$ observe that the only compact almost complex manifold supporting a circle action with discrete fixed point set is the\nsphere, since such surface must have positive Euler characteristic. In this case $\\k0=2$. So we must have $n\\geq 2$, and hence $\\k0\\geq 2$.\n\nThe proof of the rest is very similar to that of Proposition \\ref{cor n+1}, but we include it here for the sake of completeness.\nIf $N_0=0$ then all the claims in Proposition \\ref{cor n} follow from Corollary \\ref{bound on k0} ({\\bf ii}). \nSuppose that $N_0\\neq 0$. Then by Proposition \\ref{properties P} \\eqref{1a} we have that $\\Hi(z)\\not\\equiv 0$, and from\nTheorem \\ref{main theorem} \\eqref{H=0 even} and Corollary \\ref{extra root -k02} we have that \n$\\Hi(z)=\\beta(z+\\frac{n}{2})\\prod_{j=1}^{n-1}(z+j)$. \nIn order to determine $\\beta$ we can use \nProposition \\ref{properties P} \\eqref{1a}, obtaining \n$\\beta=\\frac{2\\,N_0}{n!}$, thus implying \\eqref{H k0=n}. The equations in \\eqref{n precise} follow easily from observing that\n$$\n\\sigma_{n-h}\\Big(1,2,\\ldots,n-1,\\frac{n}{2}\\Big)=\\sigma_{n-h}\\big(1,2,\\ldots,n-1\\big)+\\frac{n}{2}\\sigma_{n-h-1}(1,2,\\ldots,n-1)=\\left[ \\begin{array}{c} n \\\\ h \\end{array} \\right]+\\frac{n}{2}\\left[ \\begin{array}{c} n \\\\ h+1 \\end{array} \\right]\\,.\n$$\n\nIn order to prove \\eqref{c1 n} it is sufficient to consider \\eqref{n precise} with $h=n$ (or $h=n-1$).\nTo prove \\eqref{c1c22}, first of all observe that \n$$\n\\sigma_2(1,2,\\ldots,n-1,\\frac{n}{2})=\\sigma_2(1,2,\\ldots,n-1)+\\frac{n}{2}\\sigma_1(1,2,\\ldots,n-1)=\\frac{1}{24}n(n-1)(3n^2-n+2)\\,,\n$$\nwhere the last equality follows from \\eqref{sigma1} and \\eqref{sigma2}.\nThus if we take $h=n-2$ in \\eqref{n precise} we obtain\n\\begin{align*}\n\\mathsf{c}_1^{n-2}\\left(\\frac{\\mathsf{c}_1^2+\\mathsf{c}_2}{12}\\right)[\\mathsf{M}]& =N_0\\frac{2(n-2)!n^{n-2}}{n!}\\sigma_2(1,2,\\ldots,n-1,\\frac{n}{2})\\\\\n & = \\frac{N_0}{12}n^{n-2}(3n^2-n+2)\\,,\\\\\n\\end{align*}\nand the conclusion follows from \\eqref{c1 n}.\n\nIn order to prove \\eqref{gen fct n}, observe that, by the above discussion, if $\\k0=n$ then $\\Hi(z)$ is either of degree $n$,\nwhich happens exactly if $N_0\\neq 0$, or it is identically zero. In the first case, by Proposition \\ref{gen fct hilbert} and Corollary \\ref{U palindrom}, $\\U(t)$ is a self-reciprocal polynomial of degree\none and $\\U(0)=N_0$, thus implying \\eqref{gen fct n}.\n\n\\end{proof}\n\nFrom Propositions \\ref{cor n+1} and \\ref{cor n} we can see that the cases $\\k0=n+1$ and $\\k0=n$ are very similar, in the sense that\nthe Hilbert polynomial $\\Hi(z)$, as well as the combinations of Chern numbers $\\mathsf{c}_1^h\\,T_{n-h}[\\mathsf{M}]$, for $h=0,\\ldots,n$, and the generating\nfunction $\\Gen(t)$ of $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$, are completely determined (see Remark \\ref{num of cds}).\n\\begin{rmk}\\label{liham}\nIn recent work Li \\cite{Li} proves that if the $2n$-dimensional manifold $\\mathsf{M}$ is symplectic, the $S^1$ action Hamiltonian and $\\chi(\\mathsf{M})=n+1$, then having $\\k0=n+1$ (resp.\\ $\\k0=n$)\nis equivalent to having the same total Chern class of ${\\mathbb{C}} P^n$ (resp.\\ of the Grassmannian of oriented planes in $\\mathbb{R}^{n+2}$ with $n$ odd) which, in turns, is equivalent to having the same integral \ncohomology ring of ${\\mathbb{C}} P^n$ (resp.\\ the Grassmannian). Thus in particular, under the above hypotheses, all the Chern numbers are `standard', i.e.\\ they agree with those of ${\\mathbb{C}} P^n$ (resp.\\ of the hyperquadric).\nThe assumption $\\chi(\\mathsf{M})=n+1$ is essential, since it implies the existence of a quasi-ample line bundle (in the sense specified in Remark \\ref{hattori rmk}) which in this case is given by the pre-quantization line bundle (see also \\cite[Proposition 7.5 (i)]{GoSa}). \n\\end{rmk}\n\n\nIn the following we analyse in details the cases $\\k0=n-1$ and $\\k0=n-2$.\nObserve that if $n=1$ the index $\\k0$ cannot be zero, since the only compact almost complex surface\nthat can be endowed with a compatible $S^1$-action with isolated fixed points is the sphere, for which $\\k0= 2$.\nSo in the next proposition it is not restrictive to assume $n\\geq 2$ for $\\k0=n-1$.\n\n\\begin{prop}[$\\k0=\\mathbf{n-1}$]\\label{k0=n-1}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space of dimension $2n\\geq 4$ with index $\\k0=n-1$.\n\\begin{itemize}\n\\item[(a)]\\label{n-1 a} If $N_0\\neq 0$ and $\\mathsf{c}_1^n[\\mathsf{M}]\\neq 0$ then \n\\begin{equation}\\label{H n-1 1}\n\\Hi(z)=\\frac{4\\,N_0}{(n-2)!\\big[(n-1)^2-4a\\big]}\\Big(z^2+(n-1)z+\\frac{(n-1)^2}{4}-a\\Big)\\prod_{j=1}^{n-2}(z+j)\\,,\n\\end{equation}\nwhere $a\\in \\mathbb{R}$ is not equal to $\\frac{(n-1)^2}{4}$. Moreover \n\\begin{equation}\\label{c1n n-1}\n\\mathsf{c}_1^n[\\mathsf{M}]=\\frac{4\\,N_0\\,n(n-1)^{n+1}}{(n-1)^2-4a}\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{c1c222}\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=\\frac{4N_0(n-1)^{n-2}}{\\big[(n-1)^2-4a\\big]}\\Big[3-12a-6n+\\frac{9}{2}n^2-2n^3+\\frac{n^4}{2}\\Big]\\,.\n\\end{equation}\n\\item[(b)]\\label{n-1 b} If $N_0\\neq 0$ and $\\mathsf{c}_1^n[\\mathsf{M}]= 0$ then\n\\begin{equation}\\label{H n-1 2}\n\\Hi(z)=\\frac{N_0}{(n-2)!}\\prod_{j=1}^{n-2}(z+j)\\,,\n\\end{equation}\nand \n\\begin{equation}\\label{c1c2 n-1}\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=12\\, N_0 (n-1)^{n-2}\\,.\n\\end{equation}\n\\end{itemize}\n\nMoreover, in \\emph{(a)} and \\emph{(b)}, the generating function of $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$ is given by \n\\begin{equation}\\label{gen fct n-1 1}\n\\Gen(t)=N_0\\frac{1+b\\,t+\\,t^2}{(1-t)^{n+1}} \n\\end{equation}\nwhere $b\\in \\mathbb{Q}$ is such that $b\\,N_0\\in {\\mathbb{Z}}$ and\n\\begin{equation}\\label{c1n n-1 b}\n\\mathsf{c}_1^n[\\mathsf{M}]=N_0(b+2)(n-1)^n\\,,\n\\end{equation}\n\\begin{equation}\\label{c1c2 n-1 b}\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=N_0(n-1)^{n-2}\\big[12+\\frac{(b+2)n(n-3)}{2}\\big]\\,.\n\\end{equation}\n(Thus case \\emph{(b)} corresponds to taking $b=-2$.)\n\n\\begin{itemize} \n\\item[(c)]\\label{n-1 c}\nIf $N_0=0$ then \n\\begin{equation}\\label{H n-1 3}\n\\Hi(z)=\\gamma \\prod_{j=0}^{n-1}(z+j)\\,,\n\\end{equation}\nwhere $\\gamma=\\frac{1}{(n-1)^n n!} \\mathsf{c}_1^n[\\mathsf{M}]$.\n\\end{itemize}\nMoreover, the generating function of $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$ is given by \n\\begin{equation}\\label{gen fct n-1 3}\n\\Gen(t)=\\gamma\\,n!\\frac{t}{(1-t)^{n+1}}\\,. \n\\end{equation}\n\\end{prop}\n\\begin{rmk}\\label{integrality a}\nObserve that the value of $a$ in \\eqref{c1n n-1} cannot be arbitrary, since the following fraction\n$$\n\\frac{4N_0\\,n(n-1)}{(n-1)^2-4a}\n$$\nmust be an integer. This follows from the fact that, modulo torsion, $\\mathsf{c}_1=(n-1)\\eta_0$ for some $\\eta_0\\in H^2(\\mathsf{M};{\\mathbb{Z}})$,\nand hence $\\frac{\\mathsf{c}_1^n[\\mathsf{M}]}{(n-1)^n}$ must be an integer.\n\\end{rmk}\n\n\nThe following corollary is a straightforward consequence of Proposition \\ref{k0=n-1}\n \\begin{corollary}\\label{pos roots n-1}\n Under the same hypotheses of Proposition \\ref{k0=n-1}, we have that:\\\\\n - If $N_0\\neq 0$ then \n \\begin{itemize}\n \\item[(1)] The roots of $\\Hi(z)$ belong to the canonical strip $\\mathcal{S}_{n-1}$ if and only if $\\mathsf{c}_1^n[\\mathsf{M}]\\geq 0$, or equivalently if and only if $b\\geq -2$.\n \\item[(2)] $\\Hi(z)$ belongs to $\\mathcal{T}_{n-1}$ if and only if $\\;\\;\\;0\\leq \\mathsf{c}_1^n[\\mathsf{M}]\\leq 4N_0 n(n-1)^{n-1}$, or equivalently if and only if $\\;\\;\\;-2\\leq b \\leq 2\\displaystyle\\frac{n+1}{n-1}$.\n \\end{itemize}\n - If $N_0=0$ then the roots of $\\Hi(z)$ do not belong to $\\mathcal{S}_{n-1}$.\n \\end{corollary}\n As a result of the analysis carried out when $\\k0=n-1$, we can strengthen Theorem \\ref{RV theorem}.\n \\begin{corollary}\\label{RV1}\n Under the same hypotheses of Proposition \\ref{k0=n-1}, assume that $N_0=1$ and $n>5$. Then $\\Hi(z)$ belongs to $\\mathcal{T}_{n-1}$ if and only if $\\U(t)$ has its roots on the unit circle. \n \\end{corollary}\n \\begin{proof}\n If $N_0=1$ then by Proposition \\ref{k0=n-1} we know that $b$ is an integer. \nIf $n>5$, from Corollary \\ref{pos roots n-1} we can see that $\\Hi(z)$ belongs to\n$\\mathcal{T}_{n-1}$ if and only if $-2\\leq b\\leq 2$. Since $b$ is an integer, for all such values of $b$ the polynomial $\\U(t)=1+bt+t^2$ has its roots on the unit circle. \n \\end{proof}\n\\begin{rmk}\\label{RV n-1}\nFor $2\\leq n\\leq 5$, we have that $2\\displaystyle\\frac{n+1}{n-1}\\geq 3$; however\nfor $b\\geq 3$, the roots of $\\U(t)$ are not on the unit circle. So for $2\\leq n\\leq 5$,\nthere may exist manifolds whose associated Hilbert polynomial belongs to $\\mathcal{T}_{n-1}$, but the corresponding $\\U(t)=1+bt+t^2$ does not have its roots on the unit circle: consider for example the Fano threefold $V_5$\nin Example \\ref{examples 6} (3), for which $b=3$ and the corresponding Hilbert polynomial is given by $\\Hi_{V_5}(z)=\\frac{1}{6}\\big[5z^2+10z+6\\big](z+1)$. \n\\end{rmk}\n\n\\begin{proof}[Proof of Proposition \\ref{k0=n-1}]\n(a) If $N_0\\neq 0$ then, by Proposition \\ref{properties P} \\eqref{1a} we have that $\\Hi(z)\\not\\equiv 0$. Moreover by \\eqref{Hilbert pol}, if $\\mathsf{c}_1^n[\\mathsf{M}]\\neq 0$ then $\\deg(\\Hi)=n$.\nBy Theorem \\ref{main theorem} \\eqref{H=0 even}, if $n\\geq 3$ $\\Hi(z)$ has roots $-1,-2,\\ldots, -n+2$. By Corollary \n\\ref{property roots}, the remaining two roots belong to $\\mathcal{C}_{n-1}$ and, by Proposition \\ref{properties P} \\eqref{3a},\nthey are of the form $-\\frac{n-1}{2}-x$, $-\\frac{n-1}{2}+x$. Moreover $a:=x^2\\neq \\frac{(n-1)^2}{4}$\nsince by Proposition \\ref{properties P} \\eqref{1a} and \\eqref{3a}, $\\Hi(0)=N_0$, $\\Hi(-n+1)=(-1)^nN_0$ and by assumption $N_0\\neq 0$. Thus $\\Hi(z)=\\alpha \\Big(z^2+(n-1)z+\\frac{(n-1)^2}{4}-a\\Big)\\prod_{j=1}^{n-2}(z+j)$,\nwhere $\\alpha\\in \\mathbb{R}$ can be found by imposing $\\Hi(0)=N_0$, obtaining \\eqref{H n-1 1}.\nEquations \\eqref{c1n n-1} and \\eqref{c1c222} come from combining \\eqref{Hilbert pol} with \\eqref{H n-1 1}.\n\n\n(b) If $N_0\\neq 0$ and $\\mathsf{c}_1^n[\\mathsf{M}]=0$ then, by Proposition \\ref{properties P} \\eqref{1a} we have that $\\Hi(z)\\not\\equiv 0$\nand, by \\eqref{Hilbert pol}, $\\deg(\\Hi)\\leq n-2$.\nBy Theorem \\ref{main theorem}, if $n\\geq 3$ $\\Hi(z)$ has $n-2$ roots given by $-1,-2,\\ldots,-n+2$;\nmoreover if $n=2$ it must be a non-zero constant polynomial. Thus $\\Hi(z)$ has degree $n-2$ and it is of the form\n$\\Hi(z)=\\beta \\prod_{j=1}^{n-2}(z+j)=\\beta\\sum_{h=0}^{n-2}z^h\\sigma_{n-h-2}(1,2,\\ldots,n-2)$. By Proposition \\ref{properties P} \\eqref{1a} we have $\\beta=\\frac{N_0}{(n-2)!}$,\nand \\eqref{H n-1 2} follows.\nEquation \\eqref{c1c2 n-1} can be obtained from \\eqref{H n-1 2} and \\eqref{ah}\nby taking $h=n-2$.\n\nIn order to prove \\eqref{gen fct n-1 1} for $\\mathsf{c}_1^n[\\mathsf{M}]\\neq 0$, observe that since $\\deg(\\Hi)=n$, $N_0\\neq 0$ and $\\k0=n-1$, from Proposition \\ref{gen fct hilbert} and Corollary \\ref{U palindrom} it follows\nthat $\\U(t)=N_0(1+b\\,t+t^2)$ for some $b\\in \\mathbb{R}$. Thus we have that \n$$\n\\Gen(t)=N_0 \\frac{1+b\\,t+t^2}{(1-t)^{n+1}} = N_0\\sum_{k\\geq 0}\\left[ \\binom{n+k-2}{n}+b\\binom{n+k-1}{n}+\\binom{n+k}{n}\\right]t^k\\,, \n$$\nand by definition of $\\Gen(t)$ we have that $N_0(b+n+1)=\\Hi(1)$. Since $\\Hi(1)$ is an integer, it follows that $b\\,N_0$ must be an integer.\nMoreover, by \\eqref{H n-1 1} we have that $\\displaystyle\\frac{\\Hi(1)}{N_0}=\\frac{4(n-1)\\big[n+\\frac{(n-1)^2}{4}-a\\big]}{\\big[(n-1)^2-4a\\big]}=b+n+1$, thus obtaining $b$ in terms of $a$, and the expressions of $\\mathsf{c}_1^n[\\mathsf{M}]$ and $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$ in terms of $b$ follow from \\eqref{c1n n-1} and \\eqref{c1c222}.\n\nThe proof of \\eqref{gen fct n-1 1} when $\\mathsf{c}_1^n[\\mathsf{M}]=0$ also follows from Proposition \\ref{gen fct hilbert}, and the details are left to the reader.\n\n(c) If $N_0=0$ then, by Proposition \\ref{properties P} \\eqref{1a} and \\eqref{3a}, and Theorem \\ref{main theorem} \\eqref{H=0 even}, $\\Hi(z)$ has $n$ roots given by $0,-1,-2,\\ldots,-n+1$. If $\\mathsf{c}_1^n[\\mathsf{M}]=0$ then\nby \\eqref{Hilbert pol} and \\eqref{ah} we have that $\\deg(\\Hi)\\leq n-2$, hence $\\Hi(z)\\equiv 0$ and \\eqref{H n-1 3} follows.\nOtherwise $\\Hi(z)=\\gamma \\prod_{j=0}^{n-1}(z+j)$ where the expression for $\\gamma$ can be obtained by using\n\\eqref{Hilbert pol}, imposing that $a_n=\\gamma$.\n\nThe proof of \\eqref{gen fct n-1 3} follows easily from Proposition \\ref{gen fct hilbert}, and the details are left to the reader.\n\n\\end{proof}\n\nProposition \\ref{k0=n-1} implies that the Chern numbers $\\mathsf{c}_1^n[\\mathsf{M}]$ and $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$ are related\nby the following formula.\n\\begin{corollary}\\label{relation c122}\nUnder the same hypotheses of Proposition \\ref{k0=n-1} we have that \n$$\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]-\\frac{n(n-3)}{2(n-1)^2}\\mathsf{c}_1^n[\\mathsf{M}] = 12 N_0(n-1)^{n-2}\n$$\n\\end{corollary}\n\\begin{proof}\nWhen $N_0\\neq 0$ the claim follows from \\eqref{c1n n-1 b} and \\eqref{c1c2 n-1 b}.\n\nIf $N_0=0$ and $\\mathsf{c}_1^n[\\mathsf{M}]=0$ then from \\eqref{H n-1 3} we have $\\Hi(z)\\equiv 0$, which, by \\eqref{ah} implies that \n$$a_{n-2}=\\frac{1}{12(n-1)^{n-2}(n-2)!}\\big(\\mathsf{c}_1^n + \\mathsf{c}_1^{n-2}\\mathsf{c}_2\\big)[\\mathsf{M}]=0\\,,$$\nthus implying $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=0$, and the claim follows. \n\nOtherwise, if $N_0\\neq 0$ and $\\mathsf{c}_1^n[\\mathsf{M}]\\neq 0$, from \\eqref{H n-1 3} and \\eqref{sigma 2} we have that \n$a_{n-2}$ is \n\\begin{equation}\\label{an-2}\na_{n-2}=\\gamma \\left[ \\begin{array}{c} n \\\\ n-2 \\end{array} \\right]= \\gamma \\frac{(3n-1)n(n-1)(n-2)}{24}\\,,\n\\end{equation}\nwhere $\\gamma=\\frac{1}{(n-1)^n n!}\\mathsf{c}_1^n[\\mathsf{M}]$,\nand the claim follows from comparing the general expression of $a_{n-2}$ with \\eqref{an-2}.\n\\end{proof}\n\nAs it will be proved in Prop.\\ \\ref{dim 4}, if $(\\M,\\J,S^1)$ is an $S^1$-space of dimension $4$, the index $\\k0$ cannot be zero.\nHence it is not restrictive to assume $n\\geq 3$ for $\\k0=n-2$.\n\\begin{prop}[$\\k0=\\mathbf{n-2}$]\\label{k0=n-2}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space of dimension $2n\\geq 6$ with index $\\k0=n-2$.\n\\begin{itemize}\n\\item[(a)]\\label{n-2 a} If $N_0\\neq 0$ and $\\mathsf{c}_1^n[\\mathsf{M}]\\neq 0$ then \n\\begin{equation}\\label{H n-2 1}\n\\Hi(z)=\\frac{4\\,N_0}{(n-2)!\\big[(n-2)^2-4a\\big]}\\Big(2z+n-2\\Big)\\Big(z^2+(n-2)z+\\frac{(n-2)^2}{4}-a\\Big)\\prod_{j=1}^{n-3}(z+j)\\,,\n\\end{equation}\nwhere $a\\in \\mathbb{R}$ is not equal to $\\frac{(n-2)^2}{4}$. Moreover\n\\begin{equation}\\label{c1n n-2}\n\\mathsf{c}_1^n[\\mathsf{M}]=\\frac{8\\,N_0\\,n(n-1)(n-2)^n}{(n-2)^2-4a}\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{c1c2 n-2 2 rmk}\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=\\frac{4N_0(n-2)^{n-2}(24-24a-30n+17n^2-6n^3+n^4)}{(n-2)^2-4a}\\,. \n\\end{equation}\n\\\\\n\n\\item[(b)]\\label{n-2 b} If $N_0\\neq 0$ and $ \\mathsf{c}_1^n[\\mathsf{M}]= 0$ then\n\\begin{equation}\\label{H n-2 2}\n\\Hi(z)=\\frac{N_0}{(n-2)!}\\Big(2z+n-2\\Big)\\prod_{j=1}^{n-3}(z+j)\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{c1c2 n-2}\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=24\\, N_0 (n-2)^{n-2}\\,.\n\\end{equation}\n\\end{itemize}\nMoreover, in \\emph{(a)} and \\emph{(b)}, the generating function of $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$ is given by \n\\begin{equation}\\label{gen fct n-2}\n\\Gen(t)=N_0\\frac{1+b\\,t+b\\,t^2+t^3}{(1-t)^{n+1}} \n\\end{equation}\nwhere $b$ is such that $b\\,N_0$ is an integer and \n\\begin{equation}\\label{c1n b}\n\\mathsf{c}_1^n[\\mathsf{M}]=2N_0(b+1)(n-2)^n\\,,\n\\end{equation}\n\\begin{equation}\\label{c1c2 b}\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]=N_0(n-2)^{n-2}\\big[24+(b+1)(n-2)(n-3)\\big]\\,,\n\\end{equation}\nand case \\emph{(b)} corresponds to taking $b=-1$.\n\\begin{itemize}\n\\item[(c)]\\label{n-2 c}\nIf $N_0=0$ then \n\\begin{equation}\\label{H n-2 3}\n\\Hi(z)=\\gamma \\Big(z+\\frac{n-2}{2}\\Big)\\prod_{j=0}^{n-2}(z+j)\\,,\n\\end{equation}\nwhere $\\gamma=\\frac{1}{(n-2)^n n!}\\mathsf{c}_1^n[\\mathsf{M}]$.\n\\end{itemize}\nMoreover, the generating function of $\\{\\Hi(k)\\}_{k\\in \\mathbb{N}}$ is given by \n\\begin{equation}\\label{gen fct n-2 3}\n\\Gen(t)=\\frac{\\gamma}{2}n!\\frac{t+t^2}{(1-t)^{n+1}}\\,. \n\\end{equation}\n\n\\end{prop}\n\n\\begin{rmk}\\label{integrality a2}\nThe same comment in Remark \\ref{integrality a} applies here: the value of $a$ cannot be arbitrary, since the following fraction\n$$\n\\frac{8N_0\\,n(n-1)}{(n-2)^2-4a}\n$$\nmust be an integer.\n\\end{rmk}\n\n\nThe following corollary is very similar to Corollary \\ref{pos roots n-1}, and is a straightforward consequence of Proposition \\ref{k0=n-2}\n \\begin{corollary}\\label{pos roots n-2}\n Under the same hypotheses of Proposition \\ref{k0=n-2}, we have that:\\\\\n - If $N_0\\neq 0$ then \n \\begin{itemize}\n \\item[(1)] The roots of $\\Hi(z)$ belong to the canonical strip $\\mathcal{S}_{n-2}$ if and only if $\\mathsf{c}_1^n[\\mathsf{M}]\\geq 0$, or equivalently if and only if $b\\geq -1$.\n \\item[(2)] $\\Hi(z)$ belongs to $\\mathcal{T}_{n-2}$ if and only if $\\;\\;\\;0\\leq \\mathsf{c}_1^n[\\mathsf{M}]\\leq 8N_0 n(n-1)(n-2)^{n-2}$, or equivalently if and only if $\\;\\;\\;-1\\leq b \\leq \\displaystyle\\frac{3n^2-4}{(n-2)^2}$.\n \\end{itemize}\n - If $N_0=0$ then the roots of $\\Hi(z)$ do not belong to $\\mathcal{S}_{n-2}$.\n \\end{corollary}\nIn analogy with Corollary \\ref{RV1}, we have the following:\n\\begin{corollary}\\label{RV2}\n Under the same hypotheses of Proposition \\ref{k0=n-2}, assume that $N_0=1$ and $n>14$. Then $\\Hi(z)$ belongs to $\\mathcal{T}_{n-2}$ if and only if $\\U(t)$ has its roots on the unit circle. \n \\end{corollary}\n\\begin{proof}\nIf $N_0=1$ then by Proposition \\ref{k0=n-2}, we know that $b$ is an integer. If $n>14$, from Corollary \\ref{pos roots n-2} we can see that $\\Hi(z)$ belong to\n$\\mathcal{T}_{n-2}$ if and only if $-1\\leq b\\leq 3$. Since $b$ is an integer, for all such values of $b$ the polynomial $\\U(t)=1+bt+bt^2+t^3$ has its roots on the unit circle.\n\\end{proof}\n\\begin{rmk}\\label{RV n-2}\nFor $3\\leq n\\leq 14$, we have that $\\displaystyle\\frac{3n^2-4}{(n-2)^2}\\geq 4$; however\nfor $b\\geq 4$, the roots of $\\U(t)=1+bt+bt^2+t^3$ are not on the unit circle. In conclusion, we can say that for $3\\leq n\\leq 14$,\nthere may exist manifolds whose associated Hilbert polynomial belongs to $\\mathcal{T}_{n-2}$, but the corresponding $\\U(t)$ does not have its roots on the unit circle: consider for example the Fano threefold $V_{22}$\nin Example \\ref{examples 6-1} (2), for which $b=10$ and the corresponding Hilbert polynomial is given by $\\Hi_{V_{22}}(z)=\\frac{1}{6}\\big[11z^2+11z+6\\big](2z+1)$.\n\\end{rmk}\n\n\\begin{proof}[Proof of Proposition \\ref{k0=n-2}]\n\nThe proof of this Proposition is very similar to that of Proposition \\ref{k0=n-1}, and here we only sketch the first part.\n(a) If $N_0\\neq 0$ then, by Proposition \\ref{properties P} \\eqref{1a} we have that $\\Hi(z)\\not\\equiv 0$. Moreover by \\eqref{Hilbert pol}, if $ \\mathsf{c}_1^n[\\mathsf{M}]\\neq 0$ then $\\deg(\\Hi)=n$.\nBy Theorem \\ref{main theorem} \\eqref{H=0 even}, for $n\\geq 4$ $\\Hi(z)$ has roots $-1,-2,\\ldots, -n+3$. By Corollary \\ref{extra root -k02} one of the remaining three roots is $-\\frac{n-2}{2}$. By Corollary \\ref{property roots} the remaining two roots are on $\\mathcal{C}_{n-2}$, and\nby Proposition \\ref{properties P} \\eqref{3a} they are of the form $-\\frac{n-2}{2}-x$, $-\\frac{n-2}{2}+x$, for some $x\\in \\mathbb{R}$.\nMoreover $a:=x^2\\neq \\frac{(n-2)^2}{4}$\nsince by Proposition \\ref{properties P} \\eqref{1a} and \\eqref{3a}, $\\Hi(0)=N_0$, $\\Hi(-n+2)=(-1)^nN_0$ and by assumption $N_0\\neq 0$. \nIt follows that the Hilbert polynomial is of the form\n$$\n\\Hi(z)=\\alpha \\Big(2z+n-2\\Big)\\Big(z^2+(n-2)z+\\frac{(n-2)^2}{4}-a\\Big)\\prod_{j=1}^{n-3}(z+j)\\,,\n$$\nwhere $\\alpha$ can be found by imposing $\\Hi(0)=N_0$, thus obtaining \\eqref{H n-2 1}. \nThe rest of the proof is left to the reader. \n\\end{proof}\n\nSimilarly to the case $\\k0=n-1$, Proposition \\ref{k0=n-2} implies that the Chern numbers $\\mathsf{c}_1^n[\\mathsf{M}]$ and\n$\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$ are related by the following formula.\n\\begin{corollary}\\label{relation c122 2}\nUnder the same hypotheses of Proposition \\ref{k0=n-2} we have that \n$$\n\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]-\\frac{n-3}{2(n-2)}\\mathsf{c}_1^n[\\mathsf{M}]= 24 N_0 (n-2)^{n-2}\n$$\n\\end{corollary}\n\\begin{proof}\nThe proof of this Corollary is very similar to that of Corollary \\ref{relation c122}, and the details are left to the reader.\n\\end{proof}\n\nAs a consequence of the analysis of $\\Hi(z)$ when the index $\\k0$ is $n-2$ or $n$ we have the following\n\\begin{corollary}\\label{cor even chern classes}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space with $N_0\\neq 0$. Assume the index satisfies either $\\k0=n$, or $\\k0=n-2$ and $n\\geq 3$.\nThen the Chern numbers $\\mathsf{c}_1^n[\\mathsf{M}]$ and $\\mathsf{c}_1^{n-2}\\mathsf{c}_2[\\mathsf{M}]$ are always \\emph{even}.\n\\end{corollary}\n\\begin{proof}\nWhen $\\k0=n$ the claim follows from Proposition \\ref{cor n} \\eqref{c1 n} and \\eqref{c1c22},\nand when $\\k0=n-2$ it follows from Proposition \\ref{k0=n-2} \\eqref{c1n b} and \\eqref{c1c2 b}.\n\\end{proof}\n\n\nThe case in which $\\k0={\\bf n-3}$, where $n\\geq 4$, is not analysed in details here. However we would like to make some remarks about it when $N_0\\neq 0$\nand $\\deg(\\Hi)=n$, i.e.\\;$ \\mathsf{c}_1^n[\\mathsf{M}]\\neq 0$.\nFirst of all, observe that this is the first case in which the roots of $\\Hi(z)$ may not belong to $\\mathcal{C}_{\\k0}$ (see Corollary \\ref{position roots}). From Theorem \\ref{main theorem} \\eqref{H=0 even}, the roots of $\\Hi(z)$ are $-1,-2,\\ldots,-n+4$ (if $n>4$), plus four additional roots $z_1,z_2,z_3,z_4$. If the remaining four roots don't belong to $\\mathcal{C}_{\\k0}$, \nfrom the properties of $\\Hi(z)$ they must be of the form $-\\frac{n-3}{2}\\pm a\\pm {\\bf i}\\, b$, for some $a,b\\in \\mathbb{R}\\setminus \\{0\\}$, thus obtaining that\n\\begin{equation}\\label{k0=n-3}\n\\Hi(z)=\\alpha \\prod \\Big(z+\\frac{n-3}{2}\\pm a\\pm {\\bf i}\\,b\\Big)\\prod_{j=1}^{n-4}(z+j)\\,.\n\\end{equation}\nFrom the expression of $a_n$ in \\eqref{ah} and Proposition \\ref{properties P} \\eqref{1a} it follows that \n\\begin{equation}\\label{cassini}\n\\Big[\\Big(\\frac{n-3}{2}-a\\Big)^2+b^2\\Big]\\Big[\\Big(\\frac{n-3}{2}+a\\Big)^2+b^2\\Big]=\\frac{N_0\\,n!\\,(n-3)^n}{\\,(n-4)!\\,\\mathsf{c}_1^n[\\mathsf{M}]}\\,,\n\\end{equation}\nwhich implies that $\\mathsf{c}_1^n[\\mathsf{M}]>0$. Moreover, for a fixed value of $\\mathsf{c}_1^n[\\mathsf{M}]$, the four roots $z_1,\\ldots,z_4$ \nbelong to the \\emph{Cassini oval}\\footnote{We recall that a \\emph{Cassini oval} is a quartic plane curve given by the locus of points in $\\mathbb{R}^2\\simeq {\\mathbb{C}}$ satisfying the equation $$\\mathrm{d}(p,q_1)\\,\\mathrm{d}(p,q_2)=d^2\\,,$$ where $d\\neq 0$. The points $q_1$ and $q_2$ are called the \\emph{foci} of the Cassini oval.} of equation\n\\begin{equation}\\label{cassini eq}\n\\mathrm{d}(p,0)\\,\\mathrm{d}(p,-n+3)=\\sqrt{\\frac{N_0\\,n!\\,(n-3)^n}{\\,(n-4)!\\,\\mathsf{c}_1^n[\\mathsf{M}]}}\n\\end{equation}\nwhere $\\mathrm{d}(p,q)$ denotes the Euclidean distance from $p$ to $q$, with $p,q\\in\\mathbb{R}^2\\simeq {\\mathbb{C}}$, and the foci of this oval are the points $0$ and $-n+3$ (see Figure \\ref{Fig:cassini}).\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=\\textwidth\n\\leavevmode\n\\includegraphics[width=3.5in]{Cassini-final}\n\\end{center}\n\\caption{Examples of \\emph{Cassini ovals} of equation $\\mathrm{d}(p,q_1)\\,\\mathrm{d}(p,q_2)=d^2$ with foci $q_1=(0,0)$ and $q_2=(-4,0)$ for different values of $d$.\nThe curve passing through the origin is called the \\emph{lemniscate of Bernoulli}, and is obtained for $d=4$.}\n\\label{Fig:cassini}\n\\end{figure}\n\n\n\\medskip\n\\subsection{Conclusions on Hamiltonian and non-Hamiltonian actions.}\nAs an application of the results obtained before, we conclude the section with the proof of Theorem \\ref{nHam-char}. Observe that for $n=1$ and $n=2$ there do not exist symplectic non-Hamiltonian circle actions with nonempty discrete fixed point sets: for $n=1$ the only compact surface admitting such a symplectic circle action is a sphere, hence the action is Hamiltonian; for $n=2$ the assertion was proved by McDuff in \\cite[Proposition 2]{MD1}.\n\\begin{proof}[Proof of Theorem \\ref{nHam-char}]\nWe recall that in the symplectic case $N_0$ can be either $0$ or $1$, and it is $0$ exactly if the action is non-Hamiltonian (see\nLemma \\ref{N0 1}). \nThen the claims in (I) follow from Corollary \\ref{bound on k0 s} ({\\bf i'}), those\nin (II) and (III) from Propositions \\ref{cor n+1} and \\ref{cor n}, and those in (IV) and (V) from Corollaries \\ref{relation c122} and \\ref{relation c122 2}.\n\\end{proof}\n\\begin{rmk}\\label{other comb}\nObserve that \nby Propositions \\ref{cor n+1} and \\ref{cor n}, when $\\k0=n+1$ or $\\k0=n$ the action is Hamiltonian if and only if \\emph{all} the combinations of \nChern numbers $\\mathsf{c}_1^h\\,T_{n-h}[\\mathsf{M}]$ do not vanish, for $h=0,\\ldots,n$.\n\\end{rmk}\n\\section{Examples: low dimensions of $(\\mathsf{M},\\mathsf{J})$}\\label{examples}\n\nIn this section, we study some consequences of the results previously obtained for\n$n\\leq 4$.\nIn particular we prove that when $\\k0=n$ or $n+1$ then \\emph{all the Chern numbers of $(\\mathsf{M},\\mathsf{J},S^1)$ can be expressed as a linear combination of the $N_j$'s}, where $N_j$ denotes the number of fixed points with exactly $j$ negative weights. In the Hamiltonian category, this amounts to saying that \\emph{all the Chern numbers of $(\\mathsf{M},\\omega, S^1)$ can be expressed as linear combinations of the Betti numbers of }$\\mathsf{M}$ (see \\eqref{bi=Ni}).\n\nThe most obvious Chern number that can always be written in terms of the $N_j$'s \nis $\\mathsf{c}_n[\\mathsf{M}]$. In fact, by definition of the $N_j$'s and $\\mathsf{c}_n[\\mathsf{M}]=|\\mathsf{M}^{S^1}|$, we have \n\\begin{equation}\\label{eq cn}\n\\mathsf{c}_n[\\mathsf{M}]=\\sum_{j=0}^n N_j\\,.\n\\end{equation}\nIn \\cite{GoSa}, Godinho and the author proved that the Chern number $\\mathsf{c}_1\\mathsf{c}_{n-1}[\\mathsf{M}]$ can also be expressed in terms of the $N_j$'s. \nWe recall its explicit expression in the following\n\\begin{theorem}[\\cite{GoSa} Theorem 1.2]\\label{nostro}\n Let $(\\mathsf{M},\\mathsf{J}, S^1)$ and $N_j$ be as above. Then \n \\begin{equation}\\label{c1cn-1}\n \\mathsf{c}_1\\mathsf{c}_{n-1}[\\mathsf{M}]=\\sum_{j=0}^n N_j \\Big[6j(j-1)+\\frac{5n-3n^2}{2}\\Big]\\,.\n \\end{equation}\n\n\\end{theorem}\n\nSuppose that $(\\M,\\J,S^1)$ is an $S^1$-space\nof (real) dimension $2$. As also observed before, since we are requiring isolated fixed points,\nsuch a space must be a $2$-sphere, obtaining \n $\\k0=2$, $\\Hi(z)=1+z$ and $\\mathsf{c}_1[S^2]=2$. \n\n\\subsection{$\\mathbf{\\dim(\\mathsf{M})=4}$} First of all, observe that by \\eqref{NiN} and \\eqref{eq cn} we have\n\\begin{equation}\\label{c2 dim 4}\n \\mathsf{c}_2[\\mathsf{M}]=2N_0+N_1\\,.\n\\end{equation}\nMoreover, \nby \\eqref{NiN} and Theorem \\ref{nostro} \\eqref{c1cn-1}, for $n=2$ it follows that\n\\begin{equation}\\label{c1 n=2}\n\\mathsf{c}_1^2[\\mathsf{M}]=10 N_0 - N_1\\,.\n\\end{equation}\nThus in dimension $4$ all the Chern numbers can be expressed as a linear combination of the $N_j$'s (independently on $\\k0$).\n\\begin{rmk}\\label{pos 1}\nObserve that the necessary condition $\\mathsf{c}_1^2+\\mathsf{c}_2[\\mathsf{M}]\\equiv 0 \\mod{12}$, which must hold for\nany compact almost complex manifold, for $S^1$-spaces becomes \n$\\mathsf{c}_1^2+\\mathsf{c}_2[\\mathsf{M}]=12 N_0$ (it is equivalent to saying that the Todd genus is $N_0$). Hence, \nfor $(\\M,\\J,S^1)$, the combination of Chern numbers $\\mathsf{c}_1^2+\\mathsf{c}_2[\\mathsf{M}]$ must be a \\emph{non-negative} multiple of $12$.\n\\end{rmk}\nThe following corollary is an easy consequence of the results obtained before, applied to the symplectic category:\n\\begin{corollary}\\label{geo s}\nLet $(\\mathsf{M},\\omega)$ be a compact, connected symplectic manifold of dimension $4$ that can be endowed with a symplectic circle action with isolated fixed points.\nThen \n\\begin{equation}\\label{c1 n2 h}\n(\\mathsf{c}_1^2[\\mathsf{M}], \\mathsf{c}_2[\\mathsf{M}])=(10-b_2(\\mathsf{M}), 2+ b_2(\\mathsf{M}))\\,.\n\\end{equation}\nMoreover, any pair of integers $(p,q)$ satisfying $p+q=12$ and $p\\leq 9$ can be realized as the pair of Chern numbers $(\\mathsf{c}_1^2[\\mathsf{M}], \\mathsf{c}_2[\\mathsf{M}])$\nof a compact, connected symplectic manifold $\\mathsf{M}$ of dimension $4$ supporting a symplectic circle action with isolated fixed points.\n\\end{corollary}\n\\begin{proof}\nGiven $(\\mathsf{M},\\omega,S^1)$ of dimension $4$, a theorem of McDuff \\cite{MD1} implies that the action is Hamiltonian, and as a consequence of \\eqref{c1 n=2}, Lemma \\ref{N0 1}\nand \\eqref{bi=Ni} we obtain \\eqref{c1 n2 h}.\nThe second assertion follows from observing that $b_2(\\mathsf{M})$ is positive, it is at least one (consider $({\\mathbb{C}} P^2,\\omega_{F},S^1)$, see Example \\ref{ch}), and can be arbitrarily large: to obtain\n$(\\mathsf{M},\\omega,S^1)$ with $b_2(\\mathsf{M})=k$, it is sufficient to perform $(k-1)$ times an $S^1$-equivariant blow-up on $({\\mathbb{C}} P^2,\\omega_{F},S^1)$.\n\\end{proof}\n\\begin{exm}\\label{ch}[{\\bf The complex projective space and Hirzebruch surfaces}]\nConsider ${\\mathbb{C}} P^2$ endowed with a multiple of the Fubini-Study form $\\omega_F$, and `standard' $S^1$-action, namely $S^1$ is a circle subgroup of a 2-dimensional torus $\\mathbb{T}^2$ acting on ${\\mathbb{C}} P^2$\nin a toric way. Thus the $S^1$-action is given by\n$\\alpha \\cdot [z_0:z_1:z_2]=[z_0:\\alpha^l z_1:\\alpha^{l+m}z_2]$\nfor every $\\alpha\\in S^1$ (where $l$ and $m$ are non-zero, coprime integers) it has three fixed points and is Hamiltonian. Note that the minimal Chern number of ${\\mathbb{C}} P^2$ is $3$.\nWe denote this $S^1$-space by $({\\mathbb{C}} P^2,\\lambda \\,\\omega_F,S^1)_{l,m}$, where $\\lambda \\in \\mathbb{R}_{>0}$.\n\nFor every $k\\in {\\mathbb{Z}}$, let $\\mathcal{H}_k$ be the Hirzebruch surface: $\\{([z_0:z_1:z_2],[w_1:w_2])\\in {\\mathbb{C}} P^2 \\times {\\mathbb{C}} P^1\\mid z_1\\,w_2^k=z_2\\,w_1^k\\}$, endowed\nwith symplectic form $\\widetilde{\\omega}$ induced by multiples of the Fubini-Study forms on ${\\mathbb{C}} P^2$ and ${\\mathbb{C}} P^1$. We can give each $\\mathcal{H}_k$ an $S^1$-action, defined by:\n $\\alpha\\cdot ([z_0:z_1:z_2],[w_1:w_2])=([\\alpha^l z_0:z_1:\\alpha^{k\\,m}z_2)],[w_1,\\alpha^m w_2])$, where $l$ and $m$ are non-zero, coprime integers. This action has $4$ fixed points and is Hamiltonian.\n We denote these $S^1$-spaces by $(\\mathcal{H}_k,\\widetilde{\\omega},S^1)_{l,m}$.\n Note that the minimal Chern number of $\\mathcal{H}_k$ is $1$ if $k$ is odd and $2$ if $k$ is even, and $\\mathcal{H}_k$ is respectively called an \\emph{odd} or \\emph{even} Hirzebruch surface.\n \n\\end{exm}\n\\begin{rmk}\\label{minimal spaces}\nThe examples above are exactly the \\emph{minimal spaces} obtained in the classification of $(\\mathsf{M},\\omega,S^1)$ of dimension $4$ (if the fixed point set is not discrete, there is an additional class of minimal spaces\ngiven by ${\\mathbb{C}} P^1$-bundles over Riemann surfaces of genus $g\\geq 1$), see \\cite{AH,Au} and \\cite{K}. More precisely, in \\cite{K} Karshon proves that every $(\\mathsf{M},\\omega,S^1)$ is equivariantly symplectomorphic to a symplectic\n$S^1$-space obtained from $({\\mathbb{C}} P^2,\\lambda \\,\\omega_F,S^1)_{l,m}$ or\n $(\\mathcal{H}_k,\\widetilde{\\omega},S^1)_{l,m}$ (for suitable $\\lambda, l,m,k$ as above) by a sequence of $S^1$-equivariant blow-ups at fixed points.\\footnote{Note that the blow-up of $({\\mathbb{C}} P^2,\\lambda \\,\\omega_F,S^1)_{l,m}$\n at one fixed point is an odd Hirzebruch surface.} \n \\end{rmk}\n\\begin{rmk}\\label{ci 2}\nObserve that for every $S^1$-space $(\\mathsf{M},\\omega,S^1)$ of dimension $4$ the following inequality holds:\n\\begin{equation}\\label{ci 12}\n\\mathsf{c}_1^2[\\mathsf{M}]\\leq 3\\mathsf{c}_2[\\mathsf{M}]\\,.\n\\end{equation}\nIndeed, Corollary \\ref{geo s} implies that \\eqref{ci 12} is equivalent to $b_2(\\mathsf{M})\\geq 1$.\nNote that \\eqref{ci 12} was conjectured by Van de Ven \\cite{V}\nand proved by Miyaoka \\cite{Mi}\nfor (complex) surfaces of general type. \n\nThe following question is then natural:\n\\begin{question}\\label{inac?}\nLet $(\\M,\\J,S^1)$ be an $S^1$-space. Does inequality \\eqref{ci 12} hold? \n\\end{question}\n\\end{rmk}\nBy \\eqref{c2 dim 4} and \\eqref{c1 n=2}, proving inequality \\eqref{ci 12} for $(\\M,\\J,S^1)$ is equivalent to proving that for every such space $N_0\\leq N_1$. \nThe next proposition \nimplies that the answer to question \\ref{inac?} is `yes' for all $4$-dimensional $S^1$-spaces whose index is not one.\n\n\\begin{prop}\\label{dim 4}\nLet $(\\mathsf{M},\\mathsf{J},S^1)$ be an $S^1$-space of dimension $4$, and let\n$\\k0$, $\\Hi(z)$ and the $N_j$'s be defined as before.\nThen $N_0, N_1$ and $N_2$ are all non-zero, the first Chern class $\\mathsf{c}_1$ is not a torsion element in $H^2(\\mathsf{M};{\\mathbb{Z}})$, and $\\k0\\in \\{1,2,3\\}$. Moreover \n\\begin{itemize}\n\\item[(a)]If $\\k0=3$ then\n\\begin{equation}\\label{dim4 1}\nN_0=N_1=N_2,\\;\\;\\;\\quad \\mathsf{c}_1^2[\\mathsf{M}]=9N_0\\;\\;\\;\\quad \\mbox{and}\\;\\;\\; \\quad \\Hi(z)=\\frac{N_0}{2}(z+1)(z+2).\n\\end{equation}\n\\item[(b)] If $\\k0=2$ then\n\\begin{equation}\\label{dim4 2}\n2N_0=N_1=2N_2,\\;\\;\\;\\quad \\mathsf{c}_1^2[\\mathsf{M}]=8N_0\\;\\;\\;\\quad \\mbox{and}\\;\\;\\; \\quad \\Hi(z)=N_0(z+1)^2. \n\\end{equation}\n\\end{itemize}\n$\\;$\\\\\nGiven $(\\mathsf{M},\\omega,S^1)$ of dimension $4$ we have that\n\\begin{itemize}\n \\item[(a')] $\\k0=3$ if and only if there exists $\\lambda>0$ and coprime integers $l,m$ such that $(\\mathsf{M},\\omega,S^1)$ is equivariantly symplectomorphic to $({\\mathbb{C}} P^2,\\lambda\\,\\omega_F,S^1)_{l,m}$.\n\\item[(b')] $\\k0=2$ if and only if there exists coprime integers $l,m$, an even $k\\in {\\mathbb{Z}}$ and a symplectic form $\\widetilde{\\omega}$ on $\\mathcal{H}_k$ such that $(\\mathsf{M},\\omega,S^1)$ is equivariantly symplectomorphic to $(\\mathcal{H}_k,\\widetilde{\\omega},S^1)_{l,m}$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nLet $p$ be a fixed point, and $e^{S^1}(p)\\in H_{S^1}^2(\\{p\\};{\\mathbb{Z}})={\\mathbb{Z}}[x]$ the equivariant Euler class of the normal bundle at $p$, which is simply given by $w_{1p}w_{2p}x$, where $w_{1p}$ and $w_{2p}$ are the weights of the isotropy $S^1$-action at $p$. By the ABBV formula (Thm.\\ \\ref{abbv formula}) we must have\n\\begin{equation}\\label{ABBV}\n\\sum_{p\\in \\mathsf{M}^{S^1}}\\frac{1}{e^{S^1}(p)}=1[\\mathsf{M}]=0\\,.\n\\end{equation} \nSo it follows that $\\mathsf{M}^{S^1}$ must contain points whose product of the corresponding weights is positive, as well as those for which it is negative. Thus $N_0+N_2\\neq 0$ which, together with \\eqref{NiN}, implies that $N_0$ and $N_2$ are non-zero,\n and $N_1\\neq 0$ . From Lemma \\ref{c1 N0} (a2) it follows that $\\mathsf{c}_1$ is not a torsion element in $H^2(\\mathsf{M};{\\mathbb{Z}})$, and by Corollary \\ref{bound on k0} ({\\bf i}) that $\\k0\\in \\{1,2,3\\}$. \n\n\nIf $\\k0=3$, by Proposition \\ref{cor n+1} \\eqref{c1 n+1} we have $\\mathsf{c}_1^2[\\mathsf{M}]=9 N_0$ which, together with \\eqref{c1 n=2} and \\eqref{NiN},\nimplies $N_0=N_1=N_2$. The expression for the Hilbert polynomial follows immediately from Proposition \\ref{cor n+1}.\nThe claims in (b) follow similarly by using Proposition \\ref{cor n}.\n\nSuppose that $(\\mathsf{M},\\omega)$ and the $S^1$-action are symplectic. By Lemma \\ref{N0 1} and the fact that $N_0\\neq 0$ we have that the action is Hamiltonian and $N_0=1$ (this also reproves\nMcDuff's theorem \\cite{MD1} in the case in which the fixed point set is discrete).\nObserve that blowing-up at one fixed point increases the second Betti number $b_2$ by $1$. It follows that the two families of minimal spaces in Remark \\ref{minimal spaces}\n are the only compact, connected symplectic manifolds of dimension $4$ that can be endowed with a symplectic circle action with isolated fixed points, with $b_2\\leq 2$.\nIf $\\k0=3$, from (a) and \\eqref{bi=Ni} we have that $b_0(\\mathsf{M})=b_2(\\mathsf{M})=b_4(\\mathsf{M})=1$, and (a') follows from the classification in \\cite{K}.\nIf $\\k0=2$, from (b) we have that $b_0(\\mathsf{M})=b_4(\\mathsf{M})=1$ and $b_2(\\mathsf{M})=2$, and the claim in (b') follows as well from \\cite{K}.\n\n\\end{proof}\n\\begin{rmk}\\label{not n}\n\\begin{enumerate}\n\\item Since symplectic $S^1$-spaces of dimension $4$ are completely classified, the claims in (a') and (b') also follow from the classification in \\cite{K}. However we would like to\npoint out that Proposition \\ref{dim 4} (a) and (b) implies immediately that for $\\k0=3$ the Betti numbers of $(\\mathsf{M},\\omega,S^1)$ are exactly those of ${\\mathbb{C}} P^2$, and for $\\k0=2$ they are exactly those of a Hirzebruch surface.\n\\item The numbers $\\lambda,l,m$ appearing in (a') are determined by the `Karshon graph' $\\Gamma$ associated to \n$(\\mathsf{M},\\omega,S^1)$, as described carefully in \\cite{K}; a similar conclusion holds for the case in (b').\n\\end{enumerate}\n\\end{rmk}\n\nIf $\\k0=1$, Proposition \\ref{k0=n-1} implies that the Hilbert polynomial\ndepends on the value of $ \\mathsf{c}_1^2[\\mathsf{M}]$. It is interesting to study the position of the roots of $\\Hi(z)$ in terms of \n $\\beta=\\frac{N_1}{N_0}$. Observe that, by Proposition \\ref{dim 4}, $\\beta>0$ and if the action is Hamiltonian (and the manifold is connected) then $\\beta=b_2(\\mathsf{M})$.\nFrom the definition of Hilbert polynomial of $(\\mathsf{M},\\mathsf{J})$ and \\eqref{c1 n=2} (see Proposition \\ref{k0=n-1}), it is immediate to see that \n \\begin{equation}\\label{Hil n=2 k0=1}\n \\Hi(z)=\\frac{N_0}{2}\\big[(10-\\beta)z^2+(10-\\beta)z+2\\big]\\,.\n \\end{equation}\n Thus for $\\beta\\neq 10$ the roots, which are of the form $-\\frac{1}{2}\\pm a$ with $a$ either real or pure imaginary, have the following position:\n \\begin{itemize}\n \\item for $0<\\beta<2$ or $\\beta>10$ they are real and distinct;\n \\item for $\\beta=2$ they are real and coincide;\n \\item for $2<\\beta<10$ they live on the axis $-\\frac{1}{2}+iy$, for $y\\in \\mathbb{R}\\setminus\\{0\\}$.\n \\end{itemize}\n Moreover when $\\left\\| \\mathsf{c}_1^2[\\mathsf{M}]\\right\\|\\to +\\infty$, or equivalently when $\\beta\\to +\\infty$, the roots cluster around the ``foci\" $0$ and $-1$.\n\nObserve that by Proposition \\ref{dim 4}, in the symplectic case it is impossible to have $\\k0=1$ and $\\beta=b_2(\\mathsf{M})\\leq 2$. \nMoreover, we can have manifolds with $b_2(\\mathsf{M})$ arbitrarily large; it is sufficient to blow-up ${\\mathbb{C}} P^2$ as many times\nas we want.\n\n\\subsection{$\\mathbf{\\dim(\\mathsf{M})=6}$} \nNow suppose that $\\dim(\\mathsf{M})=6$. As a consequence of \\eqref{NiN} and \\eqref{eq cn} we have that\n\\begin{equation}\\label{c3 6}\n\\mathsf{c}_3[\\mathsf{M}]=2(N_0+N_1)\\,,\n\\end{equation}\nand, as a direct consequence of Theorem \\ref{nostro}, that\n\\begin{equation}\\label{c1c2 6}\n\\mathsf{c}_1\\mathsf{c}_2[\\mathsf{M}]=24\\,N_0\\,.\n\\end{equation}\n\\begin{rmk}\\label{pos 2}\nIn dimension $6$ the congruences that must be satisfied by the Chern numbers are\n$\\mathsf{c}_1\\mathsf{c}_2[\\mathsf{M}]\\equiv 0 \\mod{24}$, and $\\mathsf{c}_1^3[\\mathsf{M}]\\equiv \\mathsf{c}_3[\\mathsf{M}]\\equiv 0 \\mod{2}$. Equations \\eqref{c3 6} and \\eqref{c1c2 6} show that \nfor $S^1$-spaces $\\mathsf{c}_1\\mathsf{c}_2[\\mathsf{M}]$ is always a \\emph{non-negative} multiple of $24$, and $\\mathsf{c}_3[\\mathsf{M}]$ a \\emph{positive} multiple of $2$. However\nour method does not give (in)equalities for $\\mathsf{c}_1^3[\\mathsf{M}]$, unless $\\k0=3,4$, see Proposition \\ref{dim 6}.\n\\end{rmk}\nThe following proposition \nfollows immediately from\nPropositions \\ref{cor n+1}, \\ref{cor n} and Lemma \\ref{N0 1}:\n\\begin{prop}[$\\mathbf{\\dim(\\mathsf{M})=6},\\;\\k0=3,4$]\\label{dim 6}\nLet $(\\mathsf{M},\\mathsf{J},S^1)$ be an $S^1$-space of dimension $6$, and let\n$\\k0$, $\\Hi(z)$ and the $N_j$'s be defined as before. \n\\begin{itemize}\n\\item[(a)] If $\\k0=4$ then \n$$\n\\mathsf{c}_1^3[\\mathsf{M}]=64 N_0\\quad\\;\\;\\;\\mbox{and}\\;\\;\\;\\quad \\Hi(z)=\\frac{N_0}{6}(z+1)(z+2)(z+3).\n$$\n\\item[(b)] If $\\k0=3$ then\n$$\n\\mathsf{c}_1^3[\\mathsf{M}]=54 N_0\\quad\\;\\;\\;\\mbox{and}\\;\\;\\;\\quad \\Hi(z)=\\frac{N_0}{6}(2z+3)(z+1)(z+2).\n$$\n\\end{itemize} \nIf we are given $(\\mathsf{M},\\omega,S^1)$ of dimension $6$ we have that: \n\\begin{itemize}\n\\item[(i)] If the action is Hamiltonian, then $\\k0=4$ implies $(\\mathsf{c}_1^3[\\mathsf{M}],\\mathsf{c}_1\\mathsf{c}_2[\\mathsf{M}])=(64,24)$, and \n$\\k0=3$ implies $(\\mathsf{c}_1^3[\\mathsf{M}],\\mathsf{c}_1\\mathsf{c}_2[\\mathsf{M}])=(54,24)$.\n\\item[(ii)] If the action is non-Hamiltonian, then for all $\\k0\\geq 3$ we have $(\\mathsf{c}_1^3[\\mathsf{M}],\\mathsf{c}_1\\mathsf{c}_2[\\mathsf{M}])=(0,0)$.\n\\end{itemize}\n\\end{prop}\n\nWhen $\\k0<3$, the Chern number $ \\mathsf{c}_1^3[\\mathsf{M}]$ and the Hilbert polynomial $\\Hi(z)$ are not determined by the index, $N_0$ and $N_1$ (see Remark \\ref{not same}).\nFor example, if $\\k0=2$ then from Proposition \\ref{k0=n-1} it follows that for $N_0\\neq 0$ and $ \\mathsf{c}_1^3[\\mathsf{M}]\\neq 0$ we have \n\\begin{equation}\\label{k0=1,2}\n\\mathsf{c}_1^3[\\mathsf{M}]=\\frac{48 N_0}{1-a}\\quad\\mbox{and}\\quad \\Hi(z)=\\frac{N_0}{1-a}\\big[z^2+2z+1-a\\big](z+1)\n\\end{equation}\nwhere $a\\neq 1$. \nThus the roots of $\\Hi(z)\/(z+1)$ are real exactly if $ \\mathsf{c}_1^3[\\mathsf{M}]\\geq 48\\,N_0\\;\\;$ or $\\;\\;\\mathsf{c}_1^3[\\mathsf{M}]<0$.\nMoreover they cluster around the ``foci\" $0$ and $-2$ exactly if $\\left\\| \\mathsf{c}_1^3[\\mathsf{M}]\\right\\|\\to +\\infty$.\n\\begin{exm}\\label{examples 6} In the following we give examples of manifolds of dimension $6$ with $\\k0=2$, together with their associated Hilbert polynomials.\n\\begin{itemize}\n\\item[(1)] \\emph{The flag variety }$\\mathcal{F}l({\\mathbb{C}}^3)=:\\mathcal{F}$. The variety of complete flags in ${\\mathbb{C}}^3$ is a compact symplectic (indeed K\\\"ahler) manifold of dimension $6$ which can be endowed with a Hamiltonian $S^1$-action with exactly $6$ fixed points; for details about the action see \\cite[Example 5.5]{GoSa} and the discussion preceding it. The reader can verify that the definition of $\\k0$ given here coincides with that of $C$ given in \\cite{GoSa}, hence \n$\\k0=2$. Moreover $\\mathsf{c}_1^3[\\mathcal{F}]=48$, and the Hilbert polynomial is $\\Hi_{\\mathcal{F}}(z)=(z+1)^3$.\n\\item[(2)] \\emph{The product of spheres $S^2\\times S^2\\times S^2=:\\mathcal{S}$}. This is a compact symplectic (indeed K\\\"ahler) manifold which can be endowed with a Hamiltonian $S^1$-action with exactly $2^3=8$ fixed points. Moreover it can be checked that\n$\\mathsf{c}_1^3[\\mathcal{S}]=48$, and the Hilbert polynomial is $\\Hi_{\\mathcal{S}}(z)=(z+1)^3$.\n\\item[(3)] \\emph{The Fano threefold $V_5$} (for details see \\cite{M, T1} or \\cite[Example 6.14]{GoSa}). This is a Fano manifold which can be endowed with a Hamiltonian $S^1$-action with exactly $4$ fixed points. The cohomology ring is given by ${\\mathbb{Z}}[x,y]\/\\langle x^2-5y,y^2 \\rangle$ (where $x$ has degree $2$, and $y$ degree $4$), $\\k0=2$ and $\\mathsf{c}_1=2x$. Thus $\\mathsf{c}_1^3[V_5]=40$,\nand the Hilbert polynomial is $\\Hi_{V_5}(z)=\\frac{1}{6}\\big[5z^2+10z+6\\big](z+1)$.\n\\item[(4)] \\emph{A non-K\\\"ahler example} $n\\mathcal{K}$. In \\cite{T1}, Tolman constructs a $6$-dimensional compact symplectic manifold which supports a Hamiltonian action of a $2$-dimensional torus $T$\nwith isolated fixed points, but does not admit any $T$-invariant K\\\"ahler structure. Moreover this action is GKM (see \\cite{GKM}), and its index $\\k0$, as well as the Chern number $\\mathsf{c}_1^3[n\\mathcal{K}]$, can be computed from its GKM graph (see \\cite{GT}, in particular Example 5.2 and Figure 1, as well as the discussion on page 27 in \\cite{GoSa}). It can be checked that in this case $\\mathsf{c}_1=4 \\tau_1 + 2 \\tau_2$, where $\\tau_i\\in H^2(\\mathsf{M};{\\mathbb{Z}})$ is the image under $r_H$ of\nthe canonical class $\\tau_i^{T}\\in H_T^2(\\mathsf{M};{\\mathbb{Z}})$ introduced in \\cite{GT}, for $i=1,2$. Since $H^2(\\mathsf{M};{\\mathbb{Z}})={\\mathbb{Z}}\\langle \\tau_1,\\tau_2 \\rangle$, we have $\\k0=2$. Moreover $\\mathsf{c}_1^3[n\\mathcal{K}]=64$ and the Hilbert polynomial is $\\Hi_{n\\mathcal{K}}(z)=\\frac{1}{3}\\big[4z^2+8z+3\\big](z+1)$.\n\\end{itemize}\n\\begin{rmk}\\label{not same}\nNotice that the flag variety in (1) and the non-K\\\"ahler example in (4) have the same index, the same Betti numbers (hence the same $N_j$'s), but different value of $\\mathsf{c}_1^3[\\mathsf{M}]$ and different Hilbert polynomial. \n\\end{rmk}\n\\end{exm}\nIf $\\k0=1$ then from Proposition \\ref{k0=n-2} it follows that for $N_0\\neq 0$ and $\\mathsf{c}_1^3[\\mathsf{M}]\\neq 0$ we have \n\\begin{equation}\\label{k0=1,2 2}\n\\mathsf{c}_1^3[\\mathsf{M}]=\\frac{48 N_0}{1-4a}\\quad\\mbox{and}\\quad \\Hi(z)=\\frac{N_0}{1-4a}\\big[4z^2+4z+1-4a\\big](2z+1)\n\\end{equation}\nwhere $a\\neq \\frac{1}{4}$. \nThus the roots of $\\Hi(z)\/(2z+1)$ are real exactly if $\\mathsf{c}_1^3[\\mathsf{M}]\\geq 48\\,N_0\\;\\;$ or $\\;\\;\\mathsf{c}_1^3[\\mathsf{M}]<0$.\nMoreover they cluster around the ``foci\" $0$ and $-2$ exactly if $\\left\\|\\mathsf{c}_1^3[\\mathsf{M}]\\right\\|\\to +\\infty$.\n\n\\begin{exm}\\label{examples 6-1}\nIn the following we give examples of manifolds of dimension $6$ with $\\k0=1$, together with their associated Hilbert polynomials.\n\\begin{itemize}\n\\item[(1)] ${\\mathbb{C}} P^1\\times {\\mathbb{C}} P^2=:\\mathcal{C}$. This is a compact symplectic (indeed K\\\"ahler) manifold which can be endowed with a Hamiltonian $S^1$-action with $6$ fixed points. Moreover $\\mathsf{c}_1^3[\\mathcal{C}]=54$, and the Hilbert polynomial is $\\Hi_{\\mathcal{C}}(z)=\\frac{1}{2}\\big[9z^2+9z+2\\big](2z+1)$.\n\\item[(2)] \\emph{The Fano threefold $V_{22}$} (for details see \\cite{M, T1} or \\cite[Example 6.14]{GoSa}). Similarly to Example \\ref{examples 6} (3), this is a Fano manifold which can be endowed with a Hamiltonian $S^1$-action with exactly $4$ fixed points. The cohomology ring is given by ${\\mathbb{Z}}[x,y]\/\\langle x^2-22y,y^2 \\rangle$ (where $x$ has degree $2$, and $y$ degree $4$), $\\k0=1$ and $\\mathsf{c}_1=x$. Thus $\\mathsf{c}_1^3[V_{22}]=22$, and the Hilbert polynomial is $\\Hi_{V_{22}}(z)=\\frac{1}{6}\\big[11z^2+11z+6\\big](2z+1)$.\n\\end{itemize}\n\\end{exm}\n\n\\subsection{$\\mathbf{\\dim(\\mathsf{M})=8}$}\nWhen $\\dim(\\mathsf{M})=8$, from \\eqref{NiN} and \\eqref{eq cn} we have that\n\\begin{equation}\\label{eq c4 8}\n \\mathsf{c}_4[\\mathsf{M}]= 2\\,N_0+2\\,N_1+N_2\\,,\n\\end{equation}\nand from Theorem \\ref{nostro}\n\\begin{equation}\\label{c1c3 8}\n \\mathsf{c}_1\\mathsf{c}_3[\\mathsf{M}]=44\\,N_0+8\\,N_1-2\\,N_2\\,.\n\\end{equation}\nAs for the remaining Chern numbers, we can use Propositions \\ref{cor n+1} and \\ref{cor n} to prove the following\n\\begin{prop}[$\\mathbf{\\dim(\\mathsf{M})=8},\\;\\k0=4,5$]\\label{dim 8}\nLet $(\\mathsf{M},\\mathsf{J},S^1)$ be an $S^1$-space of dimension $8$, and let\n$\\k0$, $\\Hi(z)$ and the $N_j$'s be defined as before. \n\\begin{itemize}\n\\item[(a)] If $\\k0=5$ then\n\\begin{equation}\\label{k0=5 8}\n \\mathsf{c}_1^4[\\mathsf{M}]=625\\,N_0\\,,\\quad \\mathsf{c}_1^2\\mathsf{c}_2[\\mathsf{M}]=250\\,N_0\\,,\\quad \\mathsf{c}_2^2[\\mathsf{M}]=101\\,N_0-2\\,N_1+N_2\\,,\n\\end{equation}\nand $\\Hi(z)= \\displaystyle\\frac{N_0}{24}\\prod_{j=1}^4(z+j)$\\,.\n\\item[(b)] If $\\k0=4$ then\n\\begin{equation}\\label{k0=4 8}\n \\mathsf{c}_1^4[\\mathsf{M}]=512\\,N_0\\,,\\quad \\mathsf{c}_1^2\\mathsf{c}_2[\\mathsf{M}]=224\\,N_0\\,,\\quad \\mathsf{c}_2^2[\\mathsf{M}]=98\\,N_0-2\\,N_1+N_2\\,,\n\\end{equation}\nand $\\Hi(z)= \\displaystyle\\frac{N_0}{12}(z+2)\\prod_{j=1}^3(z+j)$.\n\\end{itemize} \nMoreover, if $(\\mathsf{M},\\omega)$ is a connected symplectic manifold and the $S^1$-action is Hamiltonian then\n\\begin{itemize}\n\\item[(a')] if $\\k0=5$ we have\n\\begin{equation}\\label{k0=5 8 1}\n \\mathsf{c}_1^4[\\mathsf{M}]=625\\,,\\quad \\mathsf{c}_1^2\\mathsf{c}_2[\\mathsf{M}]=250\\,,\\quad \\mathsf{c}_2^2[\\mathsf{M}]=101-2\\,b_2(\\mathsf{M})+b_4(\\mathsf{M});\n\\end{equation}\n\\item[(b')] if $\\k0=4$ we have\n\\begin{equation}\\label{k0=5 8 2}\n \\mathsf{c}_1^4[\\mathsf{M}]=512\\,,\\quad \\mathsf{c}_1^2\\mathsf{c}_2[\\mathsf{M}]=224\\,,\\quad \\mathsf{c}_2^2[\\mathsf{M}]=98-2\\,b_2(\\mathsf{M})+b_4(\\mathsf{M}).\n\\end{equation}\n\\end{itemize}\nIf $(\\mathsf{M},\\omega)$ is a connected symplectic manifold and the $S^1$-action is non-Hamiltonian then for all $\\k0\\geq 4$ \n\\begin{equation}\\label{non-ham 8}\n \\mathsf{c}_1^4[\\mathsf{M}]= \\mathsf{c}_1^2\\mathsf{c}_2[\\mathsf{M}]=0\\quad\\mbox{and}\\quad \\mathsf{c}_2^2[\\mathsf{M}]=-2N_1+N_2\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nThe only claims in \\eqref{k0=5 8} and \\eqref{k0=4 8} which do not follow directly from Propositions \\ref{cor n+1} and \\ref{cor n} are the expressions of $ \\mathsf{c}_2^2[\\mathsf{M}]$ in terms of the $N_j$'s. In order to obtain them, it is sufficient to use the expression of the Todd genus given in Corollary \\ref{todd genus comp},\nwhich for $n=4$ gives\n\\begin{equation}\\label{todd 4}\n \\frac{-\\mathsf{c}_1^4+4\\mathsf{c}_1^2\\mathsf{c}_2+3\\mathsf{c}_2^2+\\mathsf{c}_1\\mathsf{c}_3-\\mathsf{c}_4}{720}[\\mathsf{M}]=N_0\\,.\n\\end{equation}\nBy combining \\eqref{todd 4} with \\eqref{c1 n+1}, \\eqref{c1c2}, \\eqref{c1 n} and \\eqref{c1c22} we obtain the desired claims.\nIn the symplectic case, all the claims follow from Lemma \\ref{N0 1}, \\eqref{bi=Ni}, Corollary \\ref{bound on k0} and \\eqref{todd 4}.\n\n\\end{proof}\nWhen $\\k0=3$ or $\\k0=2$, from Proposition \\ref{k0=n-1} and \\ref{k0=n-2} we can see that the coefficients of the Hilbert polynomial depend on the value of $ \\mathsf{c}_1^4[\\mathsf{M}]$. The following proposition exhibits the\nrelation between $ \\mathsf{c}_1^2\\mathsf{c}_2[\\mathsf{M}]$, $ \\mathsf{c}_2^2[\\mathsf{M}]$ and $ \\mathsf{c}_1^4[\\mathsf{M}]$.\n\\begin{prop}[$\\mathbf{\\dim(\\mathsf{M})=8},\\;\\k0=2,3$]\\label{dim 8 2}\nLet $(\\mathsf{M},\\mathsf{J},S^1)$ be an $S^1$-space of dimension $8$, and let\n$\\k0$, $\\Hi(z)$ and the $N_j$'s be defined as before. Then \n\\begin{itemize}\n\\item[(a)] $\\k0=3$ implies that \n\\begin{equation}\n\\label{k0=3 c1c2} \\mathsf{c}_1^2\\mathsf{c}_2[\\mathsf{M}]=108\\,N_0+\\frac{2}{9} \\mathsf{c}_1^4[\\mathsf{M}]\\,,\n\\end{equation}\nand\n\\begin{equation}\n\\label{k0=3 c22} \\mathsf{c}_2^2[\\mathsf{M}]=82\\,N_0-2\\,N_1+N_2+\\frac{1}{27} \\mathsf{c}_1^4[\\mathsf{M}]\\,.\n\\end{equation} \n\n\\item[(b)] $\\k0=2$ implies that \n\\begin{equation}\n\\label{k0=2 c1c2} \\mathsf{c}_1^2\\mathsf{c}_2[\\mathsf{M}]=96\\,N_0+\\frac{1}{4} \\mathsf{c}_1^4[\\mathsf{M}]\\,,\n\\end{equation}\nand\n\\begin{equation}\n\\label{k0=2 c22} \\mathsf{c}_2^2[\\mathsf{M}]=98\\,N_0-2\\,N_1+N_2\\,.\n\\end{equation} \n\n\\end{itemize}\n\n\\end{prop}\n\n\n\\begin{proof}\n\n(a) In order to prove \\eqref{k0=3 c1c2}, it is sufficient to use Corollary \\ref{relation c122}, and\nequation \\eqref{k0=3 c22} can be obtained by combining \\eqref{todd 4} with \\eqref{eq c4 8}, \\eqref{c1c3 8} and \\eqref{k0=3 c1c2}.\n\n(b) Equation \\eqref{k0=2 c1c2} follows from Corollary \\ref{relation c122 2}, and\n\\eqref{k0=2 c22} can be obtained by combining \\eqref{todd 4} with \\eqref{eq c4 8}, \\eqref{c1c3 8} and \\eqref{k0=2 c1c2}.\n\\end{proof}\nWe conclude this section with the following corollary:\n\\begin{corollary}\\label{c228h}\nLet $(\\mathsf{M},\\omega)$ be a compact, connected symplectic manifold of dimension $8$ that can be endowed with a Hamiltonian circle action with\nisolated fixed points. If the minimal Chern number is \\emph{even}, then \n$$\n\\mathsf{c}_2^2[\\mathsf{M}]+2\\,b_2(\\mathsf{M})=98+b_4(\\mathsf{M})\\,.\n$$\n\\end{corollary}\n\\begin{proof}\nIf $(\\mathsf{M},\\omega)$ can be endowed with a Hamiltonian circle action with isolated fixed points, then by Corollary \\ref{minimal chern ham} the minimal Chern number coincides with the index, and it can be only $1,2,3,4$ or $5$. Since it is even, the claim follows from \n\\eqref{k0=5 8 2}, \\eqref{k0=2 c22} and \\eqref{bi=Ni}.\n\\end{proof}\n\n\\begin{rmk}\\label{pos 8}\nIt is easy to check that all the necessary congruences among the Chern numbers for $n=4$ are satisfied; in particular \n$(-\\mathsf{c}_1^4+4\\mathsf{c}_1^2\\mathsf{c}_2+\\mathsf{c}_1\\mathsf{c}_3+3\\mathsf{c}_2^2-\\mathsf{c}_4)[\\mathsf{M}]$ must be a non-negative multiple of $N_0$. \nIf $\\k0\\geq 4$ then $(2\\mathsf{c}_1^4+\\mathsf{c}_1^2\\mathsf{c}_2)[\\mathsf{M}]$ must be a non-negative multiple of $12$. However, in general,\nwe cannot conclude such non-negativity results.\n\\end{rmk}\n\n\\begin{rmk}\\label{comparison}\nIt can be checked that \\eqref{k0=2 c1c2} is equivalent to equation (7.22) in \\cite{GoSa}; here $\\mathsf{M}$ is an $8$-dimensional\ncompact symplectic manifold,\nwith a Hamiltonian $S^1$-action and exactly $5$ fixed points.\nEquation (7.22) in \\cite{GoSa} is obtained by applying some results of Hattori (see \\cite{Ha}, and Corollary 7.7, Theorem 7.11 in \\cite{GoSa}) which, however, only hold \nwhenever $(\\mathsf{M},\\mathsf{J})$ possesses a fine line bundle. Moreover, the derivation of (7.22) from such results is rather complicated, as it can be seen\nfrom the proof of \\cite[Theorem 7.11]{GoSa}.\nHere we do not need to assume the existence of\na fine line bundle, and \\eqref{k0=2 c1c2} is an immediate consequence of Corollary \\ref{relation c122 2}.\n\\end{rmk}\n\nWhen $\\k0=1$ we do not obtain any restrictions on the Chern numbers (see Corollary \\ref{cor equations chern numbers}, as well as\nthe discussion on the case $\\k0=n-3$ at the end of Section \\ref{sec: values k0}). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Sec:intro} \n\nRecently, new applications have been developed to study microscopically the reactions between superfluid nuclei \\cite{Has16,Mag16}. Using the Time-Dependent Hartree-Fock-Bogoliubov (TDHFB) theory with a Gogny interaction, the reaction $^{20}$O+$^{20}$O is simulated in Ref. \\cite{Has16}. It is shown in this reaction where both fragments are superfluid that the fusion barrier depends on the initial relative gauge angle. An amplitude of $\\Delta B$=0.4 MeV is found between the maximum and minimum heights of the barrier. This difference is due to the pairing interaction between the two fragments that is either attractive or repulsive depending on the relative phase. This effect of the superfluidity is not taken into account in actual fusion model \\cite{Bac14}.\n\nFor the heavier system $^{90}$Zr+$^{90}$Zr, Magierski et al. with the FaNDF0 functional without spin-orbit interaction find a very large amplitude of $\\Delta B$=30 MeV. With the same type of calculation, for the reaction $^{44}$Ca+$^{44}$Ca, a value of $\\Delta B$=2.3 MeV is found \\cite{Sek17}. This effect is also seen on $^{120}$Sn+$^{120}$Sn \\cite{Bul17} and on asymmetric reactions $^{86}$Zr+$^{126}$Sn \\cite{Sek17b}.\n\nNevertheless, those calculations assume a semi-classical treatment of the collective variables. Indeed, the gauge angles should not be treated as a parameter of the reaction. A more elaborate method is to restore the initial symmetry in both fragments using a projection technique. A first attempt to restore the symmetry in TDHFB has been achieved recently with simplifying assumptions \\cite{Sca17} to study the Josephson effect, but this method can not be directly used to determine the fusion barrier. \n\nA simpler method to restore the symmetry is proposed in \\cite{Sca17_proc,Reg17}. It assumes an initial uniform distribution of relative gauge angles. Then, from this distribution, an ensemble of independent TDHFB trajectory is performed leading to a final distribution of the observable of interest. In a toy model, comparisons to the exact solution show that the first and second moments of the semi-classical TDHFB distributions are accurate with respect to the exact distributions. Hence, it is expected that the TDHFB may reproduce the standard deviation of the barrier distributions. However, it has to be kept in mind that the TDHFB method cannot reproduce the tunneling effect that would increase the fluctuations of the barrier distribution. More complex methods could solve the problem with a simultaneous description of the tunneling effect and the superfluidity.\nFor example, the Density-constraint-TDHFB method (that remains to be developed) based on the Density-constraint-Time-Dependent Hartree-Fock theory \\cite{Uma06} with the consideration of the pairing correlations. In the absence of more complex theory, one can still consider that the fluctuations of the barrier due to the pairing gauge angle will be convoluted to the fluctuations of the barrier due to the tunneling effect.\n\nAccording to the former TDHFB studies, it can be conjectured the following rule for fusion reactions: In reactions where both fragments are superfluid, the second order fluctuations of the fusion barrier distribution is enhanced compared to similar reactions where at least one of the fragments is not superfluid.\nThe goal of the present work is to search for the evidence of this effect with a systematic study of the fusion experimental data. \n\nSystematic studies of fusion cross section \\cite{Siw04,Wan07,Wan17}, usually use a fitting procedure to determine the main parameters of the reaction which are the barrier height, the fusion radius and the width of the barrier. \nThis method has a drawback that the final result depends on the choice of the model parametrization.. A new method is proposed and tested in order to determine those three parameters directly from the barrier distribution without assuming a parametrization of the cross sections. \n\nThe paper is organized as follows. The local regression method is tested to reduce the uncertainties on the barrier distribution in Section \\ref{Sec:Local_Regression_method}. Then, a benchmark is performed between several methods to determine the fluctuations of the barrier in Section \\ref{Sec:Bench}. A systematic analysis of the fluctuations of the barrier is done in Section \\ref{Sec:syst}. Finally, the summary is given in Section \\ref{sec:summ}\n\n\n\\section{ Local regression method }\n\\label{Sec:Local_Regression_method}\n\nThe fusion barrier distribution is defined as,\n\t\\begin{align}\n\t\tD(B) = \\left. \\frac1{\\pi R_B^2} \\frac{ d^2[ E \\sigma_{\\rm fus}(E) ] }{dE^2} \\right|_{E=B} ,\n\t\\end{align}\nwith $R_B$ the position of the barrier that is deduced from the normalisation of the barrier distribution. The second derivative is usually computed with the three-point difference formula, \n\t\\begin{align}\n\t\t \\left. \\frac{d^2(E\\sigma_{\\rm fus}(E))}{dE^2} \\right|_{E=E_2} \\simeq \\frac{E_1 \\sigma (E_1) - 2 E_2 \\sigma (E_2) + E_3 \\sigma (E_3)}{ (\\Delta E)^2 } , \\label{eq:3points}\n\t\\end{align}\nwith $E_1=E_2-\\Delta E$ and $E_3=E_2+\\Delta E$.\nThe limitation of this method is the presence of large uncertainties due to the calculation of the second derivative. These uncertainty $\\Delta D$ can be estimated at the point $E$ by\n$ \\Delta D = \\Delta\\sigma(E) \\sqrt{6} E\\sigma(E)\/(\\Delta E)^2$ \\cite{Tim98}.\nIn practice, to diminish the uncertainties, the value of $\\Delta E$ is increased. \nThis produces a smoothing of the barrier distribution. Then, structures in the barrier distribution\nsmaller than $\\Delta E$ will not be visible. It is also necessary in experiments to have a\nfixed $\\delta E$ step when the center of mass energy varies.\n\n\nThen $\\Delta E$ will be a multiple of the $\\delta E$ value.\nIn practice, with this method, a part of the information contained in the experimental data is lost because the second derivative at the point $E_2$\nis computed from the information of only three points while there can be other experimental points at the \nvicinity of $E_2$ that can bring information on the second derivative.\n\n\tFrom this statement, a new technique to calculate the second derivative using the local regression method is proposed here. The idea is to fit the experimental data around the point at energy $E$ with a polynomial function. The fitting procedure is done with a weight function, \n\t\\begin{align}\n \t\t W(E') = \\left\\{\n \t \\begin{array}{cc}\n \t 0 & \\quad |E'-E| > L \\cr\n \t ( 1 - (|E'-E|\/L)^3 )^3 & \\quad |E'-E| \\leqslant L\n \t \\end{array} \n \t \\right. ,\n \t\\end{align}\n %\n with $L$ an adjustable parameter which controls how wide is the window around a point $E$. The parameters $a_i$ of the polynomial function,\n %\n\t \\begin{align}\n\t\t f_E(x) = \\sum_{i=0}^{N} a_i x^i , \n\t \\end{align}\nare then adjusted to reproduce the experimental value of $\\sigma(E)$. Then, by making this fitting procedure for each window centered on varying energy $E$, the local regression function $F(E)=f_E(E)$ is obtained. If it is assumed that the cross section varies smoothly in the windows around the energy E, the function $F(E)$ is expected to be closer to the real $\\sigma(E) $ function than the experimental data that contains a statistical uncertainty.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{lin_ccful_cross.pdf}\n\\end{center}\n\\caption{ Simulated fusion cross section obtained with the {\\tt CCFULL} program, in linear scale (a) and logarithmic(b). \nThe original data are shown with green lines, the blue dots represent the data with a noise and the result of the local regression method $F(E)$ is shown by the red dashed lines. } \n\\label{fig:lin_ccful_cross}\n\\end{figure}\n\n\n\nTo test, this method, a fusion cross section is simulated with the program {\\tt CCFULL} \\cite{Hag99}. \nThe reaction $^{40}$Ca+$^{96}$Zr is computed with a nucleus-nucleus Woods-Saxon potential with a parameter set\n$V_0$=87.00 MeV, $r_0$=1.13 fm and $a$=0.7 fm.\n The 3$^-$ collective excitation at energy $E_3$ = 1.89 MeV\nof the $^{96}$Zr are taken into account up to three phonons with a deformation parameter $\\beta_3$ = 0.305 \nand the 3$^-$ at energy $E_3$=3.7 MeV of the $^{40}$Ca are taken into account up to three phonons with a deformation \nparameter $\\beta_3$=0.43. \nOn this data, a random error is added with an amplitude of 5\\% and 2\\%. Note that in order to describe the reaction $^{40}$Ca+$^{96}$Zr, \nit is necessary to take into account the transfer channel \\cite{Ste07,Sca15,Esb16}. Nevertheless, the goal of this calculation is not to realistically describe the fusion barrier distribution of this system but to test the method in the case of a complex barrier which has clear structure effects.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{test_barrier_ord1.pdf}\n\\end{center}\n\\caption{ Barrier distribution computed from the original {\\tt CCFULL} results with the local regression method (green solid line) and with the three-point formula (blacked dotted line).\n The results obtained from the data with noise are shown using the local regression method (red line with colored band) and the three-point formula (blue points with error bars) . The second derivative is computed with the parameter $\\Delta E=$2 MeV.\n An artificial noise of 5\\% is applied on (a) and a noise of 2\\% on (b).} \n\\label{fig:test_barrier}\n\\end{figure}\n\n\nThe local regression method with a polynomial function at first order and a parameter $L$=2 MeV \nis then applied to this data and compared to the original cross section of Fig. \\ref{fig:lin_ccful_cross}. \nThe function obtained is found to be closer to the original cross section than the simulated experimental points.\n\nFrom this function, the second derivative is computed with the three-point formula eq. \\eqref{eq:3points}.\nNote that it is still needed to use a large $\\Delta E$ to avoid the overfitting problem.\nTo estimate the uncertainties, a Monte-Carlo technique is used. \nA set of points $\\{\\sigma_i\\}$ is created, where each point is modified with a random variable $ \\sigma_i \\rightarrow \\sigma_i + \\zeta_i $ with $ \\langle \\zeta_i \\rangle = 0$ and $ \\langle \\zeta_i^2 \\rangle = \\delta_i$, with $\\delta_i$ the uncertainty on the experimental point (here the artificial error). All the $\\zeta_i$ are independent. From this sample, the barrier distribution $D(B)$ is determined. This operation is repeated $N_{rand}$ times with other random selection.\nAfter $N_{rand}$ samples, the value of $D(B)$ is computed as the average value and the uncertainty as the standard deviation for each point.\nIn this calculation, the value $N_{\\rm rand}$=100 is chosen. The result with this method is shown on\nFig. \\ref{fig:test_barrier} with two artificial noises of 5\\% and 2\\%. One can see that the local regression \nmethod is more precise than the direct three-point formula. The error bars are smaller and the average\ncurve is closer to the exact solution.\n\n\n\n\nAlso, in Fig. \\ref{fig:test_barrier} (a), on the region between 100 MeV and 105 MeV, the results of the three-point formula do not bring any information on the barrier. While with the local regression, one can see a barrier at a position close to the real one. The position and the amplitude get closer to the real one when the percentage of error is reduced (See Fig. \\ref{fig:test_barrier} (b)). Then this method, by reducing the uncertainties allows more fine analysis of the structure of the barrier from experimental cross section data (see for example \\cite{Das98,Mon17}).\n\nIn Fig. \\ref{fig:bar_comp_exp}, this method is tested on the real experimental data \\cite{Tim98} of the reaction $^{40}$Ca+$^{96}$Zr. One can see that the three-points formula induces large uncertainties while the local regression method reduces those uncertainties. Another advantage of this method is to provide a continuous function which can be integrated.\n\n\t\t\t\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{bar_comp_exp.pdf}\n\\end{center}\n\\caption{ Barrier distribution for the reaction $^{40}$Ca+$^{96}$Zr computed from the experimental cross section \\cite{Tim98} with the three-point formula (blue points with error bars) and from the local regression ( red curve and shaded area). The value of $L=\\Delta E$= 1.77 MeV is used. } \n\\label{fig:bar_comp_exp}\n\\end{figure}\t\t\t\t\n\t\t\t\t\n\n\n\n\\section{Determination of the barrier parameters }\n\nIn order to describe the fusion barrier, three parameters are defined, the centroid barrier,\n\\begin{align}\n\tB_0 \t &= \\frac{m^{B}_1}{m^{B}_0}, \\label{eq:comp_B} \n\\end{align}\nthe fusion radius, defined in order to normalize the barrier distribution,\n\\begin{align}\n\tR_B &= \\sqrt{ \\frac{m^{B}_0}{ \\pi } }, \\label{eq:comp_R_B}\n\\end{align}\nand the barrier width,\n\\begin{align}\n\t\\sigma_B \t&= \\sqrt{ \\frac{m^{B}_2}{m^{B}_0} - \\left( \\frac{m^{B}_1}{m^{B}_0} \\right)^2}. \\label{eq:definition_sigmaB}\n\\end{align}\nThese three parameters are computed from the moment of the barrier distribution, \n\\begin{align}\n m^{B}_n = \\int_0^{E_M} B^n \\left. \\frac{d^2}{dE^2}\\left( \\frac{}{} E \\sigma(E) \\right) \\right|_{E=B} dB. \\label{eq:moment}\n\\end{align}\n$E_M$ is the maximum barrier energy. This formula assumes that above the barrier $E_M$ the barrier distribution is zero.\n\n\n\n\n\\label{Sec:Bench}\n\\subsection{Calculation from the barrier distribution}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{sigma_fct_delta_E.pdf}\n\\end{center}\n\\caption{ (a) Barrier distribution computed by the local regression method for the simulated data\n with parameter $\\Delta E$ = $L$ = 1 MeV (solid and dotted lines) and 4 MeV (triangle and crosses markers) \n computed from the {\\tt CCFULL} calculation with (red dotted line and triangles) and without the collective \n excitations (blue solid line and crosses). \n(b) Fluctuations of the barrier distribution from the {\\tt CCFULL} calculation\n with (red triangles) and without the collective excitations (blue crosses) as a function of the three-point derivative parameter $\\Delta E$. A comparison is made with the integration method (eq. \\eqref{eq:sigm_int}) with (red dotted line) and without collective excitations (blue solid line) as a function of $L$. } \n\\label{fig:sigma_fct_delta_E}\n\\end{figure}\n\nIn order to determine the fluctuations of the barrier, the standard deviation of the barrier (eq. \\eqref{eq:definition_sigmaB})\nis computed, with the integration made only with the points that have a positive value of $D(B)$.\n\n\n\n\nThe difficulty of this method is that the result depends on the parameter $\\Delta E$ used to compute the barrier. \nTo show this phenomenon, the effect of the parameter $\\Delta E$ on the barrier distribution is shown in Fig. \\ref{fig:sigma_fct_delta_E}a.\nTwo test cases are shown, the first one is the same cross section as Sec \\ref{Sec:Local_Regression_method} computed with the \ncollective 3$^-$ excitations that create structures on the barrier distribution and a calculation without any collective excitation. \nThe second barrier is almost Gaussian and has small fluctuations. When the value of the $\\Delta E$ parameter increases,\nthe barrier distribution is spread, then the value of $\\sigma_B$ increase. \n\nThe obtained value of $\\sigma_B$ as a function of $\\Delta E$ is shown in Fig. \\ref{fig:sigma_fct_delta_E}b. The value needed is the asymptotic value when $\\Delta E$ tends to zero, which is difficult to attend in practice.\nIt is then not possible to determine the correct value of $\\sigma_B$ without being dependent on the parameter $\\Delta E$.\nNote that in practice, it is also difficult to determine the maximum energy $E_M$.\n\n\n\\subsection{Integral method}\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{meth_integr.pdf}\n\\end{center}\n\\caption{$^{40}$Ca+$^{96}$Zr {\\tt CCFULL} fusion cross section multiplied by the energy (red solid line). The function $g(E)$ is shown with a dashed black line. The shaded area represents the integral of eq. \\eqref{eq:sigm_int}.} \n\\label{fig:meth_integr}\n\\end{figure}\n\n\n\n\nOne can avoid the calculation of the second derivative and then avoid the problem of convolution found in the previous section by using partial integration on eq. \\eqref{eq:moment},\n\\begin{align}\nm^{B}_0 &= \\left. \\frac{d}{dE}\\left( \\frac{}{} E \\sigma(E) \\right) \\right|_{E=E_M}, \\\\\nm^{B}_1 &= E_M( m^{B}_0 - \\sigma(E_M)), \\\\\nm^{B}_2 &= E_M^2 ( m^{B}_0 - 2 \\sigma(E_M)) + 2 \\int_0^{E_M} E \\sigma(E) dE.\n\\end{align}\nFrom which simple expressions of the main parameters of the barrier are deduced, \n\\begin{align}\nR_B^2 &= \\frac1{\\pi} \\left. \\frac{d}{dE}\\left( \\frac{}{} E \\sigma(E) \\right) \\right|_{E=E_M}, \\\\\nB_0 \t\t &= E_M \\left( 1 - \\frac{ \\sigma(E_M) }{ \\pi R_B^2} \\right), \\\\\n\\sigma_B^2 &= \\frac{2}{ \\pi R_B^2 } \\int_0^{E_M} \\left( E \\sigma(E) - g(E) \\right) dE, \\label{eq:sigm_int}\n\\end{align}\n with\n\\begin{align} \n \t\t g(E) = \\left\\{\n \t \\begin{array}{cc}\n \t 0 & \\quad E \\leqslant B_0 \\cr\n \t \\pi R_B^2 ( E -B_0 ) & \\quad E>B_0 \\cr\n \t \\end{array} \n \t \\right. .\n\\end{align}\n\n\n\n\n\nThis method requires computing the derivative of the fusion cross section at the energy $E_M$ and one integral. The integral is computed from the local regression function $F(E)$. In practice, the function $g(E)$ is first adjusted to the experimental curve (see Fig. \\ref{fig:meth_integr}) around the point $E_M$, and then the integral of Eq. \\eqref{eq:sigm_int} is computed from the local regression function $F(E)$. Note that this method is close to the one of Ref. \\cite{Das04} to compute the centroid of the barrier distribution $B_0$.\n\n\n\n\n\nUsing this method on the {\\tt CCFULL} cross section, the values of the barrier fluctuations are $\\sigma_B$ = 4.18 MeV and $\\sigma_B$ = 1.03 MeV respectively with and without excitations. Those values are very stable with the $L$ parameter as shown in Fig. \\ref{fig:sigma_fct_delta_E}b. As one can expect from the Fig. \\ref{fig:meth_integr}, the area between the two curves is very dependent on the slope of the g(E) function. \nThis method is then limited to the experimental data where the slope above the barrier can be well determined. Note that the fitting method should also be very dependent on the slope above the barrier, but, it will not be explicit in the fitting procedure. In case of data without a clear slope above the barrier, the fitting procedure will extrapolate from the cross section data below the barrier. This extrapolation, if the barrier is more complicated than the fitting function will not be accurate.\n\n\n\n\nSeveral examples of applications of this method are shown in Fig. \\ref{fig:cross}. To determine the uncertainties the same Monte-Carlo method than for the barrier is used. For each of those examples, the linear $g(E)$ function can be adjusted to the experimental data without ambiguity. In this panel of 6 cross sections, the uncertainties on the values of the fluctuations of the barrier vary from 1\\% in the case where the quality of the experimental cross section is very good ($^{40}$Ca+$^{96}$Zr) to 4\\% where the number of points is less important and the uncertainties larger ($^{40}$Ca+$^{90}$Zr).\n\n\n\n\n\n\\subsection{Fitting procedure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{comp_fit_Swi.pdf}\n\\end{center}\n\\caption{$^{40}$Ca+$^{96}$Zr {\\tt CCFULL} fusion cross section calculation with (red triangles) and without (blue crosses) collective excitation. The function eq. \\eqref{eq:fct_fit_swi} is adjusted to those cross sections and shown respectively, with a red dotted line and a blue solid line. } \n\\label{fig:comp_fit_Swi}\n\\end{figure}\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{cross.pdf}\n\\end{center}\n\\caption{ Example of the application of the integration method for several reactions. The black dashed line represents the $g(E)$ function and the red cross the experimental data. The experimental data are taken from Refs. \\cite{Tim98,Sca00,Mor00,Mor99,Jia12}. } \n\\label{fig:cross}\n\\end{figure*}\n\n\nAnother method to determine the parameters of the barrier is to fit the experimental data with a parametrization of fusion cross section \\cite{Swi05},\n\\begin{align}\n\t\\sigma_{\\rm fus} = \\pi R_B^2 \\frac{\\sigma_B}{E\\sqrt{2\\pi}} [X\\sqrt{\\pi} (1+ {\\rm erf} X ) + \\exp(-X^2)],\\label{eq:fct_fit_swi}\n\\end{align}\nwith $X=\\frac{E-B_{0}}{\\sqrt{2}\\sigma_B}$.\nThe parametrization of the fusion cross section corresponds to a Gaussian barrier distribution\nwith standard deviation $\\sigma_B$. The parameters of this function are adjusted on the fusion\ncross section obtained with the {\\tt CCFULL} program. In the case with the excitations, the parameters\nare $R_B$=11.47 fm, $B_0$=93.66 MeV and $\\sigma_B$=2.08 MeV. In the case\nwhere the excitations are not taken into account $R_B$=12.85 fm, $B_0$=100.4 MeV\nand $\\sigma_B$=1.18 MeV.\n\nIn the second case, the value of $\\sigma_B$ is very close to the one in Fig. \\ref{fig:sigma_fct_delta_E} in the limit of $\\Delta E$ small. In the case of a single barrier almost gaussian, the two methods give the same result. But with the cross section generated with structure effects, the barrier is no more Gaussian. Then the fit underestimates a lot the barrier fluctuations, $\\sigma_B$=2.08 MeV instead of about 4.5 MeV with the direct calculation. \n\nTo go beyond this approach, the fusion cross section is fitted with a sum of two functions of Eq. \\eqref{eq:fct_fit_swi} which is equivalent to assume that the barrier is composed of a sum of two Gaussians. Then the barrier width is determined by Eq. \\eqref{eq:definition_sigmaB}.\nWith this method, the barrier fluctuations are of 3.44 MeV.\n This result is closer to the correct value, but still underestimate the real fluctuations of the barrier. Note that the interesting method of the Bayesian spectral deconvolution \\cite{Hag16} could improve the present fitting procedure, but seems to be too complex to be used for a systematic analysis.\n\n\n\n\n\n\n\n\\section{Systematic analysis}\n\\label{Sec:syst}\n\n\nThe two methods (fitting procedure with two Gaussians and integral method) have been systematically applied to a large number of experimental data from the database \\cite{nrv}. 115 reactions have been selected on those data for which the slope above the barrier can be reasonably well determined. The main selection has been done on the uncertainties of the results. Only systems for which the uncertainties on the value of $\\sigma_B$ is lower than 0.75 MeV have been analyzed.\nA comparison between the results obtained by both methods is shown in Fig. \\ref{fig:comp_fit_integral}. A good agreement is found between the two methods. For 73\\% of the reactions, the two methods are giving results with a difference of less than 0.5 MeV.\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{comp_fit_integral.pdf}\n\\end{center}\n\\caption{ Fluctuations of the barrier $\\sigma_B$ determined with the integral method as a function of the fit method. } \n\\label{fig:comp_fit_integral}\n\\end{figure}\n\n\nIn order to analyze the data, I define the parameter $S$ that reflects the superfluidity of the reaction.\nFor one reaction, this parameter is computed as follow: starting with $S=0$; if the $N_1$ AND $N_2$ are non magic $S$ is changed to 1; then, if the $Z_1$ AND $Z_2$ are non magic $S$ is incremented by 1. $N_1$ and $N_2$ are the neutron numbers of the two nuclei. $Z_1$ and $Z_2$ are the proton numbers. The magic number taken here are $\\{ 8, 20, 28, 50, 82, 126 \\}$.\n\nThe value of $S$ can take three values 0, 1 and 2. If it is assumed that only the non-magic number nuclei are superfluid, then, for systems with $S=0$, no increase of the fluctuations of the barrier is expected. While with $S=1$ or $S=2$ it can be expected that the superfluidity will increase the fluctuations of the barrier and that the effect will be larger with $S=2$ where neutrons and protons of each fragment are supposed to be in the superfluid phase.\n\nIn order to not mix the superfluid effects with the fusion hindrance, only systems with $Z_1Z_2<$1500 are selected. A naive comparison of the different systems with different values of $S$ is shown in Fig. \\ref{fig:sigma_fct_z}. Where the obtained $\\sigma_B$ with the integral method is shown as a function of the parameter $z=\\frac{Z_1 Z_2}{A_1^{1\/3}+A_2^{1\/3}}$. For reactions with $z$ below 80, no effects are seen and all the reactions have small fluctuations of about 2 MeV. For systems with $z>80$ three groups can be identified, those with small $\\sigma_B$ around 2 or 3 MeV, those systems are mainly $S=0$, those with $\\sigma_B$ around 3 or 4 MeV which are mainly $S$=1 and the last group around 5 MeV is mainly composed of systems with $S=2$. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{sigma_fct_z.pdf}\n\\end{center}\n\\caption{ Fluctuations of the barrier $\\sigma_B$ determined with the integral method as a function of the $z$ parameter. } \n\\label{fig:sigma_fct_z}\n\\end{figure}\n\n\n\nThen, this first result corresponds to the expected result with the tendency $\\sigma_B^{S=2}>\\sigma_B^{S=1}>\\sigma_B^{S=0}$. Nevertheless, this analysis neglects all the other effects that play a role in the determination of the fluctuation. In particular, the deformation that is also related to the magicity of the initial fragments.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{comp_sigma.pdf}\n\\end{center}\n\\caption{ Fluctuations of the barrier $\\sigma_B$ determined with the integral method as a function of the estimated barrier from Ref. \\cite{Siw04}. } \n\\label{fig:comp_sigma}\n\\end{figure}\n\nIn order to take into account those effects, the estimate $\\sigma_B$ is computed from the model of ref. \\cite{Siw04}. This model takes into account three sources of fluctuations of the barrier, (i) the tunneling effect, (ii) the static deformation, (iii) the vibration. Then, the total width of the barrier is computed as the convolution of these effects for each fragment (1) and (2),\n\\begin{align}\n(\\sigma_B^{\\rm Siw})^2 &= {\\sigma_{\\rm Tunnel}}^2 + \\sigma_{\\rm Static}(1)^2 + \\sigma_{\\rm Static}(2)^2 \\nonumber \\\\\n&+ \\sigma_{\\rm Vib.}(1)^2 + \\sigma_{\\rm Vib.}(2)^2 . \\label{eq:sigma_siw}\n\\end{align} \nThe formula of each of the terms are given in Ref. \\cite{Siw04}. This model is empirical and has several parameters adjusted on the experimental data on a large number of systems. For each reaction, the total width of the barrier is computed only from the input $A_1$, $A_2$, $Z_1$, $Z_2$ and the $\\beta_2$ of each of the fragments. The $\\beta_2$ values are taken from the M\\\"oller table \\cite{Mol95}.\n\n\n\nBecause this last model does not take into account the effect of the superfluidity, it is expected for the systems with S=1 or S=2 that \nthe empirical model will under-estimate the fluctuations of the barrier ($\\sigma_{\\rm exp.} > \\sigma_{\\rm Siw.}$).\nA comparison between the experimental values of the fluctuations of the barrier and the obtained values from the empirical model is made in Fig. \\ref{fig:comp_sigma}. \n\n\n From this comparison, one can observe the following. i) On average, the Siwek-Wilczynska model underestimates the width of the barrier. This is due to the tendency of the fitting procedure used in Ref. \\cite{Siw04} to underestimate the barrier width and to the larger number of reactions studied here.\n ii) The experimental fluctuations of the barrier are in the range of 0 to 6 MeV. There is no system that is compatible with very large fluctuations of the order of 10 MeV. iii) A clear effect of the superfluidity is found in several reactions with $S=$1 or 2 which are found to have a larger barrier width than the expected value from the Siwek-Wilczynska model and from the general trend of systems with $S$=0.\n\n\n\n\n\\begin{table}[h]\n\\caption{ Systems with $S$=1 or 2 where an enhancement of the fluctuations of the barrier more than 1 MeV is found. The values of the $\\sigma$ are given in MeV. The type of the experiment evaporated residue (EvR) or fusion-fission (FF) done is shown in the last column. }\n\\centering \\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\\hline\nReaction & $\\;S\\:$ &$\\sigma_{\\rm Siw.}$ & $\\sigma^{\\rm integ.}_{\\rm exp.}$ & $\\;$ Ref. $\\;$ & exp. \\\\\n \\hline\n$^{40}$Ar+$^{144}$Sm\t& 1 & 2.31 & 4.39 $\\pm$ 0.44 & \\cite{Rei85} & EvR+FF \\\\\n$^{32}$S+$^{138}$Ba \t& 1 & 2.06 & 3.11 $\\pm$ 0.35 & \\cite{Gil95} & EvR+FF \\\\\n$^{40}$Ar+$^{122}$Sn \t& 1 & 1.94 & 3.41 $\\pm$ 0.44 & \\cite{Rei85} & EvR+FF \\\\\n$^{32}$S+$^{120}$Sn \t& 1 & 1.94 & 3.39 $\\pm$ 0.52 & \\cite{Tri01} & EvR \\\\\n$^{58}$Ni+$^{94}$Zr \t& 1 & 2.59 & 4.28 $\\pm$ 0.34 & \\cite{Sca91} & EvR \\\\\n$^{58}$Ni+$^{60}$Ni \t& 1 & 1.87 & 3.94 $\\pm$ 0.12 & \\cite{Ste95b} & EvR \\\\\n$^{19}$F+$^{93}$Nb \t& 1 & 1.44 & 3.23 $\\pm$ 0.25 & \\cite{Pra96} & EvR \\\\\n$^{40}$Ar+$^{154}$Sm \t& 2 & 4.27 & 5.28 $\\pm$ 0.23 & \\cite{Rei85} & EvR+FF \\\\\n$^{40}$Ar+$^{148}$Sm \t& 2 & 3.15 & 4.79 $\\pm$ 0.37 & \\cite{Rei85} & EvR+FF \\\\\n$^{32}$S+$^{110}$Pd \t& 2 & 2.65 & 4.69 $\\pm$ 0.09 & \\cite{Ste95} & EvR \\\\\n$^{40}$Ar+$^{110}$Pd \t& 2 & 2.90 & 4.62 $\\pm$ 0.73 & \\cite{Jah82} & EvR \\\\\n$^{32}$S+$^{96}$Zr \t& 2 & 2.46 & 4.35 $\\pm$ 0.05 & \\cite{Zha10} & EvR \\\\\n$^{32}$S+$^{94}$Zr \t& 2 & 1.79 & 3.34 $\\pm$ 0.08 & \\cite{Jia14} & EvR \\\\\n$^{28}$Si+$^{178}$Hf \t& 2 & 4.11 & 5.22 $\\pm$ 0.18 & \\cite{But02} & EvR+FF \\\\\n$^{28}$Si+$^{92}$Zr \t& 2 & 1.68 & 2.77 $\\pm$ 0.07 & \\cite{New01} & EvR \\\\\n \\hline\\hline\n\\end{tabular}\n\\label{Tab:S2_value_inc}\n\\end{table}\n\n\n \nThe Tab. \\ref{Tab:S2_value_inc} presents systems with S=1 or 2 that have larger fluctuations of the barrier than the estimated value from the model. The table is given here, in order to guide the future microscopic applications of TDHFB or other models that aim to quantitatively reproduce the effect of the superfluidity on the barrier.\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{comp_sigma_fit.pdf}\n\\end{center}\n\\caption{ Same as Fig. \\ref{fig:comp_sigma} with the $\\sigma_{\\exp}$ determined with the fitting procedure. } \n\\label{fig:comp_sigma_fit_alpha}\n\\end{figure}\n\nIn order to confirm the results of the Fig. \\ref{fig:comp_sigma}, the same analysis is done with the fitting method in Fig. \\ref{fig:comp_sigma_fit_alpha}. The results of the fitting method are expected to be of lower quality, but the method is more tolerant of the quality and quantity of points in the experimental data. Then, this systematic analysis includes 194 reactions. Those results are shown to confirm the enhancement of the fluctuations of the barrier for systems where $S$=1 or 2. Note that, the points with $S$=0 which presents a large width of the barrier are not present in the Fig. \\ref{fig:comp_sigma} because they have too large uncertainties.\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width= \\linewidth]{R_B_fctA13.pdf}\n\\end{center}\n\\caption{ Experimental fusion radius computed as eq. \\eqref{eq:comp_R_B}. The solid line represent the function $R_B=1.30 (A_1^{1\/3}+A_2^{1\/3})$. } \n\\label{fig:R_B_fctA13}\n\\end{figure}\n\n\nTo finish this empirical analysis, the effect of the superfluidity on the fusion radius and on the centroid of the barrier distribution is investigated. \nIn fig. \\ref{fig:comp_sigma_fit_alpha}, the fusion radii of all the selected reactions, including the systems where it is expected an effect of the fusion hindrance ($Z_1Z_2>$1500) are shown. Those last reactions do not follow the general trend $R_B\\simeq 1.3 (A_1^{1\/3}+A_2^{1\/3}) $ and present a small radius in the range $3