Iterated residues and Bernoulli sums
by
András Szenes
MIT

We review recent work on computation of multidimensional sums using iterated residues. Consider the ring of rational functions with poles along the elements of some real central hyperplane arrangement. The goal is to study invariantly defined functionals on this ring. It turns out that there is a well defined notion of the "constant" term of a function in this ring, which leads to a more conceptual understanding of broken-circuit bases via iterated residues. If one fixes a compatible lattice of full rank, then one can consider the sum of the values of a rational function on the regular part of the lattice (a Bernoulli sum). In the multidimensional context, such sums first appeared in the work of Witten on 2-dimensional gauge theory. In a certain sense, this summation is again an invariant functional, which is closely related to the constant term. This allows one to write explicit formulas for these Bernoulli sums.

Date received: June 4, 1999