On positivity of Ehrhart polynomials

TL;DR

This paper surveys various families of integral polytopes that are known or conjectured to have positive Ehrhart coefficients, exploring the reasons behind their positivity and presenting examples with negative coefficients.

Contribution

It provides a comprehensive overview of Ehrhart positivity across different polytope families and discusses underlying reasons and open questions.

Findings

Identifies polytope families with proven Ehrhart positivity

Provides examples of polytopes with negative Ehrhart coefficients

Presents conjectures and open questions about Ehrhart positivity

Abstract

Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.