Explicit and efficient formulas for the lattice point count in rational polygons using Dedekind-Rademacher sums

Details Explicit and efficient formulas for the lattice point count in rational polygons using Dedekind-Rademacher sums Explicit and efficient formulas for the lattice point count in rational polygons using Dedekind-Rademacher sums

2001-11-30

arxiv.org/abs/math/0111329v2

Discrete & Comp. Geom. 27 (2002), 443--459

Mathematics

Combinatorics

16 pages, updated journal reference

Scientific paper

We give explicit, polynomial-time computable formulas for the number of integer points in any two-dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind-Rademacher sums, which are polynomial-time computable finite Fourier series. As a by-product we rederive a reciprocity law for these sums due to Gessel, which generalizes the reciprocity law for the classical Dedekind sums. In addition, our approach shows that Gessel's reciprocity law is a special case of the one for Dedekind-Rademacher sums, due to Rademacher.

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