Editing Stanley's reciprocity theorem (section)

== Definitions ==
A '''rational cone''' is the set of all ''d''-[[tuple]]s

:(''a''<sub>1</sub>, ..., ''a''<sub>''d''</sub>) 

of [[nonnegative integer]]s satisfying a [[system of inequalities]]

:<math>M\left[\begin{matrix}a_1 \\  \vdots  \\ a_d\end{matrix}\right] \geq \left[\begin{matrix}0 \\  \vdots \\  0\end{matrix}\right]</math>

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

:<math>F(x_1,\dots,x_d)=\sum_{(a_1,\dots,a_d)\in {\rm cone}} x_1^{a_1}\cdots x_d^{a_d}.</math>

The generating function ''F''<sub>int</sub>(''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>) 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 function]]s.