A seminal result of E. Ehrhart states that the number of integer lattice points in the dilation of a rational polytope by a positive integer $k$ is a quasi-polynomial function of $k$ — that is, a “polynomial” in which the coefficients are themselves periodic functions of $k$. Using a result of F. Liu on the Ehrhart polynomials of cyclic polytopes, we construct not-necessarily-convex rational polytopes of arbitrary dimension in which the periods of the coefficient functions appearing in the Ehrhart quasi-polynomial take on arbitrary values.

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具有任意周期Ehrhart系数的有理多面体

E. Ehrhart的一项开创性结果指出，在有理多边形的正整数扩张中，整数晶格点的数量 $ķ$ 是...的拟多项式函数 $ķ$ —即“多项式”，其中系数本身是 $ķ$。使用F. Liu关于循环多边形的Ehrhart多项式的结果，我们构造了任意维的不必要凸的有理多边形，其中出现在Ehrhart拟多项式上的系数函数的周期取任意值。