Classifying rational polygons with small denominator and few interior lattice points

TL;DR

This paper introduces algorithms for classifying rational polygons with fixed denominator and interior lattice points, generating a large dataset of diverse polygons by describing maximal polygons and their subpolygons with redundancy elimination.

Contribution

The paper presents a novel classification method for rational polygons with fixed parameters, including a systematic approach to generate and eliminate redundant cases.

Findings

Generated a dataset of billions of polygons

Successfully classified diverse rational polygons

Developed algorithms for maximal and subpolygon enumeration

Abstract

We present algorithms for classifying rational polygons with fixed denominator and number of interior lattice points. Our approach is to first describe maximal polygons and then compute all subpolygons, where we eliminate redundancy by a suitable normal form. Executing our classification, we obtained a dataset of several billions of polygons covering a wide variety of cases.