2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62781We 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.16 pages, updated journal referenceCombinatoricsNumber Theory05A15, 52C05, 11H06, 11L03text