Ehrhart f*-coefficients of polytopal complexes are non-negative integers

Abstract

The Ehrhart polynomial LP of an integral polytope P counts the number of integer points in integral dilates of P. Ehrhart polynomials of polytopes are often described in terms of their Ehrhart h*-vector (aka Ehrhart δ-vector), which is the vector of coefficients of LP with respect to a certain binomial basis and which coincides with the h-vector of a regular unimodular triangulation of P (if one exists). One important result by Stanley about h*-vectors of polytopes is that their entries are always non-negative. However, recent combinatorial applications of Ehrhart theory give rise to polytopal complexes with h*-vectors that have negative entries. In this article we introduce the Ehrhart f*-vector of polytopes or, more generally, of polytopal complexes K. These are again coefficient vectors of LKwith respect to a certain binomial basis of the space of polynomials and they have the property that the f*-vector of a unimodular simplicial complex coincides with its f-vector. The main result of this article is a counting interpretation for the f*-coefficients which implies that f*-coefficients of integral polytopal complexes are always non-negative integers. This holds even if the polytopal complex does not have a unimodular triangulation and if its h*-vector does have negative entries. Our main technical tool is a new partition of the set of lattice points in a simplicial cone into discrete cones. Further results include a complete characterization of Ehrhart polynomials of integral partial polytopal complexes and a non-negativity theorem for the f*-vectors of rational polytopal complexes.