06026273

j

06026273 Chen, Sheng Li, Nan Sam, Steven V. Generalized Ehrhart polynomials. Trans. Am. Math. Soc. 364, No. 1, 551-569 (2012). 2012 American Mathematical Society, Providence, RI EN Ehrhart polynomials integer solutions of Diophantine equations

doi: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 $nP$ is a quasi-polynomial in $n$. The authors generalize this theorem by allowing the vertices of $nP$ to be arbitrary rational functions in $n$. They show that the number of lattice points in $nP$ is a quasi-polynomial for sufficiently large $n$. Further the authors show that this problem is closely related to Ehrhart's conjecture on the solution-counting problem for a parametrized linear Diophantine equation with coefficients polynomial in $n$. Oleg Karpenkov (Graz)