Department of Mathematics

Copyright © 2016 University College London. We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449-1466]. By well-known decompositions, it is sufficient to consider the case of affine cones s+c, where is an arbitrary real vertex and is a rational polyhedral cone. For a given rational subspace L, we define the intermediate generating functions SL(s+c)(ξ) by integrating an exponential function over all lattice slices of the affine cone s+c parallel to the subspace L and summing up the integrals. We expose the bidegree structure in parameters and ξ, which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math. 12 (2012), 435-469] and [Intermediate sums on polyhedra: computation and real Ehrhart theory. Mathematika 59 (2013), 1-22]. The bidegree structure is key to a new proof for the Baldoni-Berline-Vergne approximation theorem for discrete generating functions [Local Euler-Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes. Contemp. Math. 452 (2008), 15-33], using the Fourier analysis with respect to the parameter and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.