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

TL;DR

This paper classifies all quadratic packing polynomials on rational sectors of the plane, providing a complete characterization of such polynomials that bijectively map lattice points to natural numbers.

Contribution

It offers a complete classification of quadratic packing polynomials on rational sectors, extending the understanding of polynomial bijections on lattice points.

Findings

All quadratic packing polynomials on rational sectors are characterized.

The classification covers all such polynomials explicitly.

The results generalize previous work on packing polynomials in specific regions.

Abstract

A polynomial $p(x,y)$ on a region $S$ in the plane is called a packing polynomial if the restriction of $p(x,y)$ to $S\cap \mathbb{Z}^2$ yields a bijection to $\mathbb{N}$. In this paper, we determine all quadratic packing polynomials on rational sectors of $\mathbb{R}^2$.