Interlacing Ehrhart Polynomials of Reflexive Polytopes

TL;DR

This paper proves conjectures about the zeros of Ehrhart polynomials of reflexive polytopes, demonstrating interlacing properties that confirm their zeros lie on a specific line in the complex plane, extending previous observations.

Contribution

It introduces a new interlacing method to establish zero distribution of Ehrhart polynomials for a broad class of reflexive polytopes from graphs.

Findings

Ehrhart polynomials have zeros on a specific line in the complex plane.

Interlacing property is established for these polynomials.

Confirms several conjectures about zero locations.

Abstract

It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann {\zeta} function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method is to prove a stronger property called interlacing.