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arXiv · 2607.15886

Ehrhart $h^*$-distributions

Benjamin Braun·Max Hlavacek·Cesar J. Meza·Santiago Morales·Andrés R. Vindas-Meléndez
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Abstract

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.

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Benjamin Braun, Max Hlavacek, Cesar J. Meza, Santiago Morales, Andrés R. Vindas-Meléndez. 2026-07-17. Ehrhart $h^*$-distributions. https://arxiv.org/abs/2607.15886

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