- On counting integral points in a convex rational polytope Jean B. Lasserre (lasserrelaas.fr) Eduardo S. Zeron (eszeronmath.cinvestav.mx) Abstract: Given a convex rational polytope $\Omega(b):=\{x\in\R^n_+\,|\,Ax=b\}$, we consider the function $b\mapsto f(b)$, which counts the nonnegative integral points of $\Omega(b)$. A closed form expression of its $\Z$-transform $z\mapsto \mathcal{F}(z)$ is easily obtained so that $f(b)$ can be computed as the inverse $\Z$-transform of $\mathcal{F}$. We then provide two variants of an inversion algorithm. As a by-product, one of the algorithms provides the Ehrhart polynomial of a convex integer polytope $\Omega$. We also provide an alternative that avoids the complex integration of $\mathcal{F}(z)$ and whose main computational effort is to solve a linear system. This latter approach is particularly attractive for relatively small values of $m$, where $m$ is the number of nontrivial constraints (or rows of $A$) Keywords: Category 1: Combinatorial Optimization (Polyhedra ) Category 2: Integer Programming (Other ) Citation: Math. Oper. Res. 28 (2003), pp. 853--870. Download: Entry Submitted: 03/02/2003Entry Accepted: 03/03/2003Entry Last Modified: 10/05/2004Modify/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.