Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra

This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the unweighted case (i.e., h = 1). In contrast to Barvinok's method, our method is local, obtains an approximation on the level of generating functions, handles the general weighted case, and provides the coefficients in closed form as step polynomials of the dilation. To demonstrate the practicality of our approach we report on computational experiments which show even our simple implementation can compete with state of the art software.Comment: 34 pages, 2 figure

INTERMEDIATE SUMS ON POLYHEDRA: COMPUTATION AND REAL EHRHART THEORY

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvi-nok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. For a given semi-rational polytope p and a rational subspace L, we integrate a given polyno-mial function h over all lattice slices of the polytope p parallel to the subspace L and sum up the integrals. We first develop an al-gorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomi-als. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory

INTERMEDIATE SUMS ON POLYHEDRA II:BIDEGREE AND POISSON FORMULA

Abstract. We continue our study of intermediate sums over polyhedra,interpolating between integrals and discrete sums, whichwere introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to considerthe case of affine cones s+c, where s is an arbitrary real vertex andc is a rational polyhedral cone. For a given rational subspace L,we integrate a given polynomial function h over all lattice slicesof the affine cone s + c parallel to the subspace L and sum up theintegrals. We study these intermediate sums by means of the intermediategenerating functions SL(s+c)(ξ), and expose the bidegreestructure in parameters s and ξ, which was implicitly used in thealgorithms in our papers [Computation of the highest coefficients ofweighted Ehrhart quasi-polynomials of rational polyhedra, Found.Comput. Math. 12 (2012), 435–469] and [Intermediate sums onpolyhedra: Computation and real Ehrhart theory, Mathematika 59(2013), 1–22]. The bidegree structure is key to a new proof for theBaldoni–Berline–Vergne approximation theorem for discrete generatingfunctions [Local Euler–Maclaurin expansion of Barvinokvaluations and Ehrhart coefficients of rational polytopes, Contemp.Math. 452 (2008), 15–33], using the Fourier analysis with respectto the parameter s and a continuity argument. Our study alsoenables a forthcoming paper, in which we study intermediate sumsover multi-parameter families of polytopes