Counting lattice points of rational polyhedra

Abstract

The generating function F(P)=∑α∈P∩ZNxα for a rational polytope P carries all essential information of P. In this paper we show that for any positive integer n, the generating function F(P, n) of nP=nx:x∈P can be written as F(P, n)=∑α∈APα(n)xnα, where A is the set of all vertices of P and each Pα(n) is a certain periodic function of n. The Ehrhart reciprocity law follows automatically from the above formula. We also present a formula for the coefficients of Ehrhart polynomials in terms of elementary symmetric functions. © 2000 Academic Press.