Classification of Quadratic Packing Polynomials on Sectors of $\mathbb{R}^2$

TL;DR

This paper classifies all quadratic packing polynomials that bijectively map lattice points in rational sectors of the plane onto non-negative integers, extending previous results in the field.

Contribution

It provides a complete classification of quadratic packing polynomials on rational sectors, generalizing earlier partial results.

Findings

All quadratic packing polynomials on rational sectors are characterized.

The classification extends known results to broader classes of sectors.

The work generalizes previous theorems by Stanton, Nathanson, and Fueter and Pólya.

Abstract

We study quadratic polynomials giving bijections from the integer lattice points of sectors of $\mathbb{R}^2$ onto $\mathbb{N}_0$ , called packing polynomials. We determine all quadratic packing polynomials on rational sectors. This generalizes results of Stanton, Nathanson, and Fueter and P\'olya.