Generalized Ehrhart polynomials

Details Generalized Ehrhart polynomials Generalized Ehrhart polynomials

2010-02-19

arxiv.org/abs/1002.3658v2

Trans. Amer. Math. Soc. 364 (2012), 551-569

Mathematics

Combinatorics

18 pages, no figures; v2: Sections 4 and 5 added, proofs and exposition have been expanded and clarified

Scientific paper

10.1090/S0002-9947-2011-05494-2

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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