Explicit Formula for Counting Lattice Points of Polyhedra

Given $z\in C^n$ and $A\in Z^{m\times n}$, we consider the problem of evaluating the counting function $h(y;z):=\sum\{z^x : x\in Z^n; Ax=y, x\geq 0\}$. We provide an explicit expression for $h(y;z)$ as well as an algorithm with possibly numerous but very simple calculations. In addition, we exhibit finitely many fixed convex cones, explicitly and exclusively defined by $A$, such that for any $y\in Z^m$, the sum $h(y;z)$ can be obtained by a simple formula involving the evaluation of $\sum z^x$ over the integral points of those cones only. At last, we also provide an alternative (and different) formula from a decomposition of the generating function into simpler rational fractions, easy to invert.