Abstract
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.
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Lasserre, J.B., Zeron, E.S. (2007). Simple Explicit Formula for Counting Lattice Points of Polyhedra. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_28
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DOI: https://doi.org/10.1007/978-3-540-72792-7_28
Publisher Name: Springer, Berlin, Heidelberg
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