Examples and counterexamples in Ehrhart theory

TL;DR

This paper explores inequalities and properties of Ehrhart and $h^*$-polynomials, including positivity, unimodality, and real-rootedness, providing new results, counterexamples, and connections within algebraic combinatorics.

Contribution

It introduces new inequalities, constructs counterexamples to unimodality conjectures, and links Ehrhart positivity with $h^*$-real-rootedness, advancing understanding in Ehrhart theory.

Findings

Constructed polytopes with pathological Ehrhart properties

Disproved variations of the unimodality conjecture for IDP polytopes

Established a connection between Ehrhart positivity and $h^*$-real-rootedness

Abstract

This article provides a comprehensive exposition about inequalities that the coefficients of Ehrhart polynomials and $h^*$-polynomials satisfy under various assumptions. We pay particular attention to the properties of Ehrhart positivity as well as unimodality, log-concavity and real-rootedness for $h^*$-polynomials. We survey inequalities that arise when the polytope has different normality properties. We include statements previously unknown in the Ehrhart theory setting, as well as some original contributions in this topic. We address numerous variations of the conjecture asserting that IDP polytopes have a unimodal $h^*$-polynomial, and construct concrete examples that show that these variations of the conjecture are false. Explicit emphasis is put on polytopes arising within algebraic combinatorics. Furthermore, we describe and construct polytopes having pathological properties on their Ehrhart coefficients and roots, and we indicate for the first time a connection between the notions of Ehrhart positivity and $h^*$-real-rootedness. We investigate the log-concavity of the sequence of evaluations of an Ehrhart polynomial at the non-negative integers. We conjecture that IDP polytopes have a log-concave Ehrhart series. Many additional problems and challenges are proposed.