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

In this paper we prove that for all degree $6$ polynomials with rational coefficients that $F({\mathbb{Z}}^2)\mathrm{\ne }{\mathbb{Z}}_{\ge 0}$. The answers a question of B. Poonen and J. S. Lew in the degree 6 case. This work builds on previous work with S. Yamagishi.