2003/06/02 by Matthias Beck
Mathematics · #math.CO #msc:05A15 #msc:11D75
published as Electronic J. Comb. 6, no. 1 (1999), R 37 · 9 pages
arxiv created 2003/06/02 · arxiv updated 2009/11/30
We generalize Ehrhart's idea 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 "vector-dilated" simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes. 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.