Counting lattice points of rational polyhedra (Q1590968)

Let \(L(P,n)\) be the number of lattice points in the dilated set \(nP\) of a bounded set \(P \subset \mathbb{R}^N\) for any integer \(n \geq 1\) and the standard lattice \(\mathbb{Z}^N\). \textit{E. Ehrhart} [J. Reine Angew. Math. 226, 1-29; ibid. 227, 25-49 (1967; Zbl 0155.37503)] proved that \(L(P,n)\) is a polynomial of \(n\) for lattice polytopes \(P\). This result extends to rational polytopes \(P\), i.e., to convex hulls of finite sets of points in \(\mathbb{R}^N\) with rational coordinates. The authors show an analogous extension for the generating function \(F(P,n)\) of the different approach to counting lattice points in polytopes by \textit{M. Brion} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No. 4, 653-663 (1988; Zbl 0667.52011)]. Further, they present a closed formula for \(L(P,n)\) and \(L(P^0,n)\) of a rational simplex \(P\) with the relative interior \(P^0\), and a formula for the coefficients of Ehrhart polynomials in terms of elementary symmetric functions.