Stanley's reciprocity theorem

Stanley's reciprocity theorem, named after the mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the integer-point generating function of a rational cone and the generating function of the cone's interior.

Definitions

A rational cone is a subset of <math>{\bf R}^d</math> consisting of all points satisfying a finite set of homogeneous linear inequalities with integer coefficients or, alternatively, the nonnegative span of a finite set of integer vectors. That is, a rational cone has the two alternative descriptions

:<math>C = \left\{ x \in {\bf R}^d : Ax \le 0 \right\}</math>

for some <math>m \times d</math> integer matrix (i.e., is defined by the halfspaces given by the rows of ), and

:<math>C = \left\{ By : y \ge 0 \right\}</math>

for some <math>d \times n</math> integer matrix (i.e., is defined as the nonnegative span of the columns of ).

The integer-point generating function (also called <i>integer-point transform</i>) of such a cone is

:<math>F_C(x_1,\dots,x_d)=\sum_{(a_1,\dots,a_d)\in C \cap {\bf Z}^d} x_1^{a_1}\cdots x_d^{a_d}.</math>

The generating function <math>F_