On the Relationship Between Ehrhart Unimodality and Ehrhart Positivity

TL;DR

This paper investigates the relationship between Ehrhart unimodality and Ehrhart positivity in lattice polytopes, demonstrating that these properties do not imply each other in dimensions greater than two, and providing new examples and insights.

Contribution

It shows that Ehrhart unimodality and positivity are independent properties in higher dimensions, with new examples and answers to open questions in Ehrhart theory.

Findings

No general implication between Ehrhart unimodality and positivity in dimensions > 2.

Constructed new examples of polytopes exhibiting each property independently.

Provided answers to open problems in Ehrhart theory.

Abstract

For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are to (1) determine if its (Ehrhart) $h^\ast$-polynomial is unimodal and (2) to determine if its Ehrhart polynomial has only positive coefficients. The former property of a lattice polytope is known as Ehrhart unimodality and the latter property is known as Ehrhart positivity. These two properties are often simultaneously conjectured to hold for interesting families of lattice polytopes, yet they are typically studied in parallel. As to answer a question posed at the 2017 Introductory Workshop to the MSRI Semester on Geometric and Topological Combinatorics, the purpose of this note is to show that there is no general implication between these two properties in any dimension greater than two. To do so, we investigate these two properties for families of well-studied lattice polytopes, assessing one property where previously only the other had been considered. Consequently, new examples of each phenomena are developed, some of which provide an answer to an open problem in the literature. The well-studied families of lattice polytopes considered include zonotopes, matroid polytopes, simplices of weighted projective spaces, empty lattice simplices, smooth polytopes, and $s$-lecture hall simplices.