Universal inequalities in Ehrhart Theory

TL;DR

This paper establishes universal inequalities among the coefficients of the $h^*$-vector of lattice polytopes, independent of their dimension or degree, revealing fundamental relations in Ehrhart theory.

Contribution

It proves the existence of universal inequalities for the $h^*$-vector coefficients that hold across all lattice polytopes, extending the understanding of Ehrhart polynomial relations.

Findings

Coefficients $h^*_1$ and $h^*_2$ satisfy Scott's inequality when $h^*_3=0$

Universal inequalities are independent of polytope dimension and degree

Relations among $h^*$-vector coefficients are established for all lattice polytopes.

Abstract

In this paper, we show the existence of universal inequalities for the $h^*$-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the $h^*$-polynomial which are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients $h^*_1$ and $h^*_2$ of the $h^*$-vector $(h^*_0,h^*_1,\ldots,h^*_d)$ of a lattice polytope of any degree satisfy Scott's inequality if $h^*_3=0$.