Ricardo Diaz

The Ehrhart polynomial of a lattice -simplex

The problem of counting the number of lattice points inside a lattice polytope in Rn has been studied from a variety of perspectives, including the recent work of Pommersheim and Kantor-Khovanskii using toric varieties and Cappell-Shaneson using Grothendieck-Riemann-Roch. Here we show that the Ehrhart polynomial of a lattice n-simplex has a simple analytical interpretation from the perspective of Fourier Analysis on the n-torus. We obtain closed forms in terms of cotangent expansions for the coefficients of the Ehrhart polynomial, that shed additional light on previous descriptions of the Ehrhart polynomial. The number of lattice points inside a convex lattice polytope in R (a polytope whose vertices have integer coordinates) has been studied intensively by combinatorialists, algebraic geometers, number theorists, Fourier analysts, and differential geometers. Algebraic geometers have shown that the Hilbert polynomials of toric varieties associated to convex lattice polytopes precisely describe the number of lattice points inside their dilates [3]. Number theorists have estimated lattice point counts inside symmetric bodies in R to get bounds on ideal norms and class numbers of number fields. Fourier analysts have estimated the number of lattice points in simplices using Poisson summation (see Siegel’s classic solution of the Minkowski problem [14] and Randol’s estimates for lattice points inside dilates of general planar regions [13]). Differential geometers have also become interested in lattice point counts in polytopes in connection with the Durfree conjecture [18]. Let Z denote the n-dimensional integer lattice in R, and let P be an ndimensional lattice polytope in R. Consider the function of an integer-valued variable t that describes the number of lattice points that lie inside the dilated polytope tP : L(P , t) = the cardinality of {tP} ∩ Z. Ehrhart [4] inaugurated the systematic study of general properties of this function by proving that it is always a polynomial in t ∈ N, and that in fact L(P , t) = Vol(P)t + 1 2 Vol(∂P)tn−1 + · · ·+ χ(P) Received by the editors August 4, 1995, and, in revised form, December 1, 1995. 1991 Mathematics Subject Classification. Primary 52B20, 52C07, 14D25, 42B10, 11P21, 11F20, 05A15; Secondary 14M25, 11H06.