Lattice points in vector-dilated polytopes

2012/04/27 by Martin Henk, Henk, Martin, Eva Linke +1
Mathematics · #11H06 #11P21 #52C07 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1204.6142

openalex publication_date 2012/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For A∈ℤ^{m× n} we investigate the behaviour of the number of lattice points in P_A(b)=\x∈ℝⁿ:Ax≤ b\, depending on the varying vector b. It is known that this number, restricted to a cone of constant combinatorial type of P_A(b), is a quasi-polynomial function if b is an integral vector. We extend this result to rational vectors b and show that the coefficients themselves are piecewise-defined polynomials. To this end, we use a theorem of McMullen on lattice points in Minkowski-sums of rational dilates of rational polytopes and take a closer look at the coefficients appearing there.

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