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

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