Generalized Ehrhart polynomials

Abstract

Let P be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations P(n) = nP is a quasi-polynomial in n. We generalize this theorem by allowing the vertices of P(n) to be arbitrary rational functions in n. In this case we prove that the number of lattice points in P(n) is a quasi-polynomial for n sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in n, and we explain how these two problems are related. © 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.