Counterexamples of the conjecture on roots of Ehrhart polynomials

TL;DR

This paper presents counterexamples to a longstanding conjecture about the location of roots of Ehrhart polynomials of integral convex polytopes, challenging previous assumptions in geometric combinatorics.

Contribution

It provides explicit counterexamples that disprove the conjecture on the roots' real parts of Ehrhart polynomials.

Findings

Counterexamples show roots can lie outside the conjectured bounds

Disproves the longstanding conjecture on Ehrhart polynomial roots

Highlights the need to revise theories on Ehrhart polynomial roots

Abstract

An outstanding conjecture on roots of Ehrhart polynomials says that all roots $\alpha$ of the Ehrhart polynomial of an integral convex polytope of dimension $d$ satisfy $-d \leq \Re(\alpha) \leq d-1$. In this paper, we suggest some counterexamples of this conjecture.