Effective lattice point counting in rational convex (original) (raw)

Simple Explicit Formula for Counting Lattice Points of Polyhedra

2007

Given z ∈ ℂn and A ∈ ℤm×n , we provide an explicit expression and an algorithm for evaluating the counting function h(y;z): = ∑ { z x | x ∈ ℤn ;Ax=y,x ≥ 0}. The algorithm only involves simple (but possibly numerous) calculations. In addition, we exhibit finitely many fixed convex cones of ℝn explicitly and exclusively defined by A, such that for any y ∈ ℤm , h(y;z) is obtained by a simple formula that evaluates ∑ 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.

Approximations of polytope enumerators using linear expansions

Several scientific problems are represented as sets of linear (or affine) con-straints over a set of variables and symbolic constants. When solutions of inter-est are integers, the number of such integer solutions is generally a meaningful information. Ehrhart polynomials are functions of the symbolic constants that count these solutions. Unfortunately, they have a complex mathematical struc-ture (resembling polynomials, hence the name), making it hard for other tools to manipulate them. Furthermore, their use may imply exponential computational complexity. This paper presents two contributions towards the useability of Ehrhart polynomials, by showing how to compute the following polynomial functions: an approximation and an upper (and a lower) bound of an Ehrhart polynomial. The computational complexity of this polynomial is less than or equal to that of * The original version of this report was submitted to the HiPEAC 2005 conference, under the title Approximating Ehrhart Polynomi...