Intermediate Sums on Polyhedra Ii:bidegree and Poisson Formula

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasipolynomial 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 s is an arbitrary real vertex and c is a rational polyhedral cone. For a given rational subspace L, we integrate a given polynomial function h over all lattice slices of the affine cone s+ c parallel to the subspace L and sum up the integrals. We study these intermediate sums by means of the intermediate generating functions S(s+c)(ξ), and expose the bidegree structure in parameters s 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 s and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.