A "rational cone" is the set of all d-tuples
of nonnegative integers satisfying a system of inequalities
where M is a matrix of integers. A d-tuple satisfying the corresponding strict inequalities, i.e., with ">" rather than "≥", is in the interior of the cone.
The generating function of such a cone is
The generating function Fint(x1, ..., xd) of the interior of the cone is defined in the same way, but one sums over d-tuples in the interior rather than in the whole cone.
It can be shown that these are rational functions. Stanley's reciprocity theorem states that
Matthias Beck, Mike Develin, and Sinai Robins have shown how to prove this by using the calculus of residues. Develin has said that this amounts to proving the result "without doing any work".