Additive number theory and inequalities in Ehrhart theory

TL;DR

This paper establishes a novel link between Ehrhart theory and additive number theory, leading to infinitely many new inequalities for the coefficients of the $h^*$-polynomial of lattice polytopes, with applications in low dimensions.

Contribution

It introduces a new approach connecting Ehrhart theory with additive number theory, producing infinitely many new inequalities for $h^*$-polynomial coefficients.

Findings

Produced infinitely many new inequalities for $h^*$-polynomial coefficients.

Deduced all balanced inequalities for polytopes with interior lattice points in dimensions up to 6.

Significantly extended the known classes of inequalities beyond previous methods.

Abstract

We introduce a powerful connection between Ehrhart theory and additive number theory, and use it to produce infinitely many new classes of inequalities between the coefficients of the $h^*$-polynomial of a lattice polytope. This greatly improves upon the three known classes of inequalities, which were proved using techniques from commutative algebra and combinatorics. As an application, we deduce all possible `balanced' inequalities between the coefficients of the $h^*$-polynomial of a lattice polytope containing an interior lattice point, in dimension at most 6.