The distribution of roots of Ehrhart polynomials for the dual of root polytopes of type C

TL;DR

This paper investigates the roots of Ehrhart polynomials for the dual of root polytopes of type C, revealing they all share the same real part and exhibit interlacing properties across dimensions.

Contribution

It proves that roots of these Ehrhart polynomials have a fixed real part and demonstrates their interlacing property across different dimensions.

Findings

Roots have real part -1/2

Ehrhart polynomials exhibit interlacing property

Results hold for all dimensions studied

Abstract

In this paper, we study the Ehrhart polynomial of the dual of the root polytope of type C of dimension $d$, denoted by $C_d^*$. We prove that the roots of the Ehrhart polynomial of $C_d^*$ have the same real part $-1/2$, and we also prove that the Ehrhart polynomials of $C_d^*$ for $d=1,2,\cdots$ has the interlacing property.