Non-injectivity of Nonzero Discriminant Polynomials and Applications to Packing Polynomials

TL;DR

This paper proves that quadratic polynomials with nonzero discriminant cannot be injective on integer lattice points within convex cones, leading to the conclusion that quadratic packing polynomials do not exist on irrational sectors of the plane.

Contribution

It establishes a fundamental non-injectivity result for quadratic polynomials with nonzero discriminant on lattice points in convex cones, with implications for packing polynomials.

Findings

Quadratic polynomials with nonzero discriminant are not injective on lattice points in convex cones.

No quadratic packing polynomials exist on irrational sectors of the plane.

The result links algebraic properties of polynomials to geometric packing problems.

Abstract

We show that an integer-valued quadratic polynomial on $\mathbb{R}^2$ can not be injective on the integer lattice points of any affine convex cone if its discriminant is nonzero. A consequence is the non-existence of quadratic packing polynomials on irrational sectors of $\mathbb{R}^2$.