A Closer Look at Lattice Points in Rational Simplices

Abstract. We generalize Ehrhart's idea ([Eh]) of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n + 1 rational vertices, we use its description as the intersection of n + 1 halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We give an elementary proof that the lattice point counts in the interior and closure of such a vectordilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes ([Ma], [Mc], [St]). As an example, we derive a lattice point count formula for a rectangular rational triangle, which enables us to compute the number of lattice points inside any rational polygon.

• Type: Journal
• Year: 1999
• Where: COMBINATORICS
• Authors: Matthias Beck
• Keywords: COMBINATORICS 1999, Dilation Factors, Lattice Points, Rational Polytopes