Ehrhart polynomials of integral simplices with prime volumes

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.