### Generalized Ehrhart polynomialsGeneralized Ehrhart polynomials

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 Author Chen, Sheng ♦ Li, Nan ♦ Sam, Steven V. Source arXiv.org Content type Text File Format PDF Date of Submission 2010-02-18 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Combinatorics ♦ Mathematics - Number Theory ♦ 52C07 ♦ 05A16 ♦ 11D45 ♦ 11D04 ♦ math 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. Description Reference: Trans. Amer. Math. Soc. 364 (2012), 551-569 Educational Use Research Learning Resource Type Article Page Count 18