# Integral points in rational polygons: a numerical semigroup approach

## TL;DR

This paper introduces a simple method using two-generator numerical semigroups to count integral points in rational polygons, starting with right-angled triangles, aiding in the broader problem of counting points in rational polygons.

## Contribution

It provides a novel elementary formula for counting integral points in rational polygons using numerical semigroups with two generators.

## Key findings

- Derived a formula for integral points in right-angled triangles with rational vertices.
- Extended the approach to general rational polygons.
- Simplified the counting process using elementary semigroup theory.

## Abstract

In this paper we use an elementary approach by using numerical semigroups (specifically, those with two generators) to give a formula for the number of integral points inside a right-angled triangle with rational vertices. This is the basic case for computing the number of integral points inside a rational (not necessarily convex) polygon.