埃尔哈特\(h^*\)分布
Ehrhart $h^*$-distributions
中文总结 AI 辅助
研究格多面体的埃尔哈特\(h^*\)分布，确定其均值、方差等，建立高阶矩与多项式系数联系，考虑实根情况获新不等式，还得出分布序列渐近正态的条件并应用于多种多面体。
中文摘要 AI 辅助
每个具有非负实系数的多项式在归一化后都会产生一个有限概率分布。格多面体\(P\)的埃尔哈特\(h^*\)多项式是一个非负整数多项式，它编码了\(P\)的正整数膨胀的整点计数。我们研究相应的有限分布，即\(h^*\)分布。我们确定了这些分布的均值和方差，建立了高阶矩与埃尔哈特多项式系数之间的联系，并研究了它们在\(d\)维概率单纯形中的聚点。我们考虑实根\(h^*\)分布的特殊情况，应用现有的尾界来获得由自反多面体产生的实根\(h^*\)多项式系数的新线性不等式。我们通过建立实根\(h^*\)分布序列渐近正态的充分条件来得出结论，并将我们的结果应用于各种多面体族，包括 zonotopes 和 Pitman-Stanley 多面体。
英文摘要
Every polynomial with real non-negative coefficients yields a finite probability distribution after normalization. The Ehrhart $h^*$-polynomial of a lattice polytope $P$ is a non-negative integer polynomial that encodes the integer-point counts for positive integer dilations of $P$. We study the corresponding finite distributions, which we call $h^*$-distributions. We determine the mean and variance of these distributions, establish a connection between higher moments and Ehrhart polynomial coefficients, and study their cluster points in the $d$-dimensional probability simplex. We consider the special case of real-rooted $h^*$-distributions, applying existing tail bounds to obtain new linear inequalities for the coefficients of real-rooted $h^*$-polynomials arising from reflexive polytopes. We conclude by establishing sufficient conditions under which a sequence of real-rooted $h^*$-distributions is asymptotically normal, and we apply our results to various families of polytopes, including zonotopes and Pitman-Stanley polytopes.
