Generalized ehrhart polynomials

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.