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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.
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Sheng Chen, Nan Li, Steven V Sam. Generalized Ehrhart polynomials. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.239-246, ⟨10.46298/dmtcs.2857⟩. ⟨hal-01186285⟩

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