Classification and Ehrhart Theory of Denominator 2 Polygons

TL;DR

This paper introduces an algorithm for classifying denominator 2 polygons with a fixed number of lattice points, analyzes their Ehrhart quasi-polynomials, and provides bounds and classifications for these polynomials.

Contribution

It presents a new algorithm for growing and enumerating denominator 2 polygons with lattice points and characterizes their Ehrhart quasi-polynomials.

Findings

Algorithm for growing denominator 2 polygons

Enumeration of polygons with few lattice points

Complete classification of Ehrhart polynomials with zero interior points

Abstract

We present an algorithm for growing the denominator $r$ polygons containing a fixed number of lattice points and enumerate such polygons containing few lattice points for small $r$. We describe the Ehrhart quasi-polynomial of a rational polygon in terms of boundary and interior point counts. Using this, we bound the coefficients of Ehrhart quasi-polynomials of denominator 2 polygons. In particular, we completely classify such polynomials in the case of zero interior points.