Residue formulae for vector partitions and Euler-MacLaurin sums

Given a finite set of vectors spanning a lattice and lying in a halfspace of a real vector space, to each vector $a$ in this vector space one can associate a polytope consisting of nonnegative linear combinations of the vectors in the set which sum up to $a$. This polytope is called the partition polytope of $a$. If $a$ is integral, this polytope contains a finite set of lattice points corresponding to nonnegative integral linear combinations. The partition polytope associated to an integral $a$ is a rational convex polytope, and any rational convex polytope can be realized canonically as a partition polytope. We consider the problem of counting the number of lattice points in partition polytopes, or, more generally, computing sums of values of exponential-polynomial functions on the lattice points in such polytopes. We give explicit formulae for these quantities using a notion of multi-dimensional residue due to Jeffrey-Kirwan. We show, in particular, that the dependence of these quantities on $a$ is exponential-polynomial on "large neighborhoods" of chambers. Our method relies on a theorem of separation of variables for the generating function, or, more generally, for periodic meromorphic functions with poles on an arrangement of affine hyperplanes.