Quartic polynomials in two variables do not represent all non-negative integers

TL;DR

This paper proves that no degree-4 polynomial with rational coefficients in two variables can take all non-negative integers as values over the integer lattice, answering a specific open question.

Contribution

It establishes a non-existence result for quartic polynomials representing all non-negative integers in two variables, resolving an open problem in the field.

Findings

No quartic polynomial in two variables over rationals maps all integer points to non-negative integers.

The result applies specifically to degree-4 polynomials, providing a definitive answer to the posed question.

This advances understanding of polynomial value sets over integer lattices.

Abstract

In this paper, we prove that there does not exist $F \in \mathbb{Q}[x,y]$ of degree $4$ such that $F(\mathbb{Z}^2) = \mathbb{Z}_{\geq 0}$. In particular, this answers a question by John S. Lew and Bjorn Poonen for quartic polynomials.