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

Matthias Beck·Sinai Robins
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Abstract

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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Matthias Beck, Sinai Robins. 2003-06-04. Explicit and efficient formulas for the lattice point count in rational polygons using Dedekind-Rademacher sums. https://arxiv.org/abs/math/0111329

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