# INTERMEDIATE SUMS ON POLYHEDRA II:BIDEGREE AND POISSON FORMULA

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## Abstract

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.

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