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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.