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Abstract: Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhartstates that the number of lattice points in the dilations $Pn = nP$ is aquasi-polynomial in $n$. We generalize this theorem by allowing the vertices ofPn to be arbitrary rational functions in $n$. In this case we prove that thenumber of lattice points in Pn is a quasi-polynomial for $n$ sufficientlylarge. Our work was motivated by a conjecture of Ehrhart on the number ofsolutions to parametrized linear Diophantine equations whose coefficients arepolynomials in $n$, and we explain how these two problems are related.

Author: Sheng Chen, Nan Li, Steven V Sam


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