- Simple Explicit Formula for Counting Lattice Points of Polyhedra J.B. Lasserre(lasserrelaas.fr) E.S. Zeron(eszeronmath.cinvestav.mx) Abstract: 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 simple computations. In addition, we exhibit finitely many fixed convex cones of $R^n$ 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. Keywords: Computational geometry; lattice polytopes. Category 1: Combinatorial Optimization (Polyhedra ) Category 2: Integer Programming (Other ) Citation: To be presented at IPCO 2007, Cornell, June 2007 Download: [PDF]Entry Submitted: 02/14/2007Entry Accepted: 02/14/2007Entry Last Modified: 02/14/2007Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.