Ehrhart polynomial roots of reflexive polytopes

TL;DR

This paper investigates the roots of Ehrhart polynomials of reflexive polytopes, characterizing specific root properties in low dimensions and providing counterexamples in higher dimensions.

Contribution

It characterizes reflexive polytopes with roots on the canonical line for dimensions up to 7 and introduces the half-strip condition, showing its validity up to dimension 5 and providing a counterexample in dimension 10.

Findings

Reflexive polytopes in dim <= 7 satisfy the canonical line hypothesis.

Reflexive polytopes in dim <= 5 satisfy the half-strip condition.

Counterexample in dim 10 violates the half-strip condition.

Abstract

Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when dim(P) <= 7. We also consider the "half-strip condition", where all roots z satisfy -dim(P)/2 <= Re(z) <= dim(P)/2-1, and show that this holds for any reflexive polytope with dim(P) <= 5. We give an example of a 10-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi--Shibata in dimension 34.