A new class of magic positive Ehrhart polynomials of reflexive polytopes

TL;DR

This paper introduces a new class of reflexive polytopes with Ehrhart polynomials that are magic positive, aiding in understanding their real-rootedness and expanding the scope of known positive cases.

Contribution

It proves the magic positivity of Ehrhart polynomials for Stasheff polytopes and partially for duals of symmetric edge polytopes of cycles, expanding the class of polytopes with this property.

Findings

Ehrhart polynomials of Stasheff polytopes are magic positive.

Partial proof of magic positivity for duals of symmetric edge polytopes of cycles.

Enhances understanding of positivity properties in Ehrhart polynomials.

Abstract

The magic positivity of Ehrhart polynomials is a useful tool for proving the real-rootedness of the $h^\ast$-polynomials. In this paper, we provide a new class of reflexive polytopes whose Ehrhart polynomials are magic positive. First, we prove that the Ehrhart polynomials of Stasheff polytopes are magic positive. Second, we provide a partial proof of the magic positivity of the Ehrhart polynomials of the dual polytopes of the symmetric edge polytopes of cycles.