H*-Polynomials with Roots on the Unit Circle

Abstract

For an n-dimensional lattice simplex with vertices given by the standard basis vectors and where has positive entries, we investigate when the Ehrhart -polynomial for factors as a product of geometric series in powers of z. Our motivation is a theorem of Rodriguez-Villegas implying that when the -polynomial of a lattice polytope P has all roots on the unit circle, then the Ehrhart polynomial of P has positive coefficients. We focus on those for which has only two or three distinct entries, providing both theoretical results and conjectures/questions motivated by experimental evidence.