Properties of Ehrhart Polynomials whose Roots Lie on the Critical Line

TL;DR

This paper investigates Ehrhart polynomials with roots on the critical line, providing improved bounds for their roots and confirming a special case of Braun's conjecture, advancing understanding of their root distribution.

Contribution

It introduces bounds for Ehrhart polynomial roots on the critical line and confirms a specific case of Braun's conjecture.

Findings

Roots of Ehrhart polynomials are bounded quadratically.

Polytopes with roots on the critical line satisfy an improved bound.

Confirmed a special case of Braun's conjecture.

Abstract

We study a class of polynomials that has all of its roots on the critical line and shares many properties with Ehrhart polynomials. Braun showed that the roots of Ehrhart polynomials are bounded quadratically and Higashitani provided examples for polytopes whose Ehrhart polynomial roots come close to this bound. In the case of polytopes which have their roots on the critical line, we present an improved bound. As a side effect this confirms a special case of a conjecture of Braun.