Ehrhart polynomials of integral simplices with prime volumes

TL;DR

This paper investigates the properties of Ehrhart polynomials of integral simplices with prime normalized volumes, establishing new relations and classifying possible δ-vectors for volumes 5 and 7.

Contribution

It introduces new equalities and inequalities for δ-vectors of simplices with prime volumes and classifies all δ-vectors for volumes 5 and 7.

Findings

Established new equalities and inequalities for δ-vectors.

Classified all δ-vectors for simplices with normalized volume 5.

Classified all δ-vectors for simplices with normalized volume 7.

Abstract

For an integral convex polytope $\Pc \subset \RR^N$ of dimension $d$, we call $\delta(\Pc)=(\delta_0, \delta_1,..., \delta_d)$ the $\delta$-vector of $\Pc$ and $\vol(\Pc)=\sum_{i=0}^d\delta_i$ its normalized volume. In this paper, we will establish the new equalities and inequalities on $\delta$-vectors for integral simplices whose normalized volumes are prime. Moreover, by using those, we will classify all the possible $\delta$-vectors of integral simplices with normalized volume 5 and 7.