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

Abstract

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

We made a few changes in the wording and fixed typos. 25 pages. See link for Poonen's question on MathOverflow: https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers