Explicit and Efficient Formulas for the Lattice Point Count in Rational Polygons Using Dedekind - Rademacher Sums

@article{Beck2002ExplicitAE,
title={Explicit and Efficient Formulas for the Lattice Point Count in Rational Polygons Using Dedekind - Rademacher Sums},
author={Matthias Beck and Sinai Robins},
journal={Discrete & Computational Geometry},
year={2002},
volume={27},
pages={443-459}
}

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… CONTINUE READING