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

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. The full beauty of the subject of generating functions emerges only from tuning in on both channels: the discrete and the continuous. Herb Wilf [W, p. vii]