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