A finite calculus approach to Ehrhart polynomials (Q976720)

To a rational polytope \(P \subset {\mathbb R}^d\) (i.e., the convex hull of finitely many points in \({\mathbb Q}^d\)), we associate the integer-point counting function \(L_P(t) := \# \left( tP \cap {\mathbb Z}^d \right)\), defined for positive integers \(t\). Ehrhart's famous theorem [\textit{E. Ehrhart}, C. R. Acad. Sci., Paris 254, 616--618 (1962; Zbl 0100.27601)] asserts that \(L_P\) is a \textit{quasipolynomial} in \(t\), i.e., \(L_P\) is of the form \[ L_P(t) = c_n(t) \, t^n + c_{ n-1 }(t) \, t^{ n-1 } + \cdots + c_1(t) \, t + c_0(t) \, , \] where \(c_0, c_1, \dots, c_n\) are periodic functions of \(t\). If \(P\) is an \textit{integral} polytope, i.e., the vertices of \(P\) are in \({\mathbb Z}^d\), then the period of \(c_0, c_1, \dots, c_n\) is one, i.e., \(L_P\) is a polynomial. The paper under review gives a new, elementary proof of Ehrhart's theorem by inductively summing over cross sections of a rational polytope. This proof is then adapted to give novel proofs of two other central theorems about Ehrhart quasipolynomials, namely Ehrhart-Macdonald reciprocity, which gives the \textit{interior} lattice-point count by evaluating \(\pm L_P(-t)\) [\textit{I. G. Macdonald}, J. Lond. Math. Soc., II. Ser. 4, 181--192 (1971; Zbl 0216.45205)], and McMullen's theorem on the periods of an Ehrhart quasipolynomial [\textit{P. McMullen}, Arch. Math. 31, 509--516 (1978; Zbl 0387.52007)].