What can be said about the polynomials $f\in\mathbb Q[x, y]$ which are nonnegative on $\mathbb R\times \mathbb R$?
Motivation: this may lead to progress in the question about polynomial onto map $\mathbb Z\times \mathbb Z\to\mathbb N$, but I post it separately as it's interesting in itself.
Note: there are examples of polynomials nonnegative on $\mathbb Z\times \mathbb Z$, but not bounded from below on $\mathbb R\times \mathbb R$, e.g. $(x^2-x)y^2$, so this doesn't apply directly.
The following theorem of Artin -- his solution of Hilbert's 17th problem, but in a stronger form than Hilbert himself asked for -- answers the question.
Theorem (Artin, 1927): Let $F$ be a subfield of $\mathbb{R}$ that has a unique ordering, and let $f(t) = f(t_1,\ldots,t_n) \in F(t_1,\ldots,t_n)$ be a rational function such that $f(a) \geq 0$ for all $a = (a_1,\ldots,a_n) \in F^n$ for which $f$ is defined. Then $f$ is a sum of squares of rational functions with coefficients in $F$.
A proof can be found in Jacobson, Basic Algebra II, Section 11.4.
Note that the tempting strengthening -- that if $f$ is a polynomial, it is a sum of squares of polynomials -- is false, as Hilbert himself showed.
Well, that's a function $\mathbb Q\times\mathbb Q\to \mathbb Q_{\geq 0}$. However, it is the same as a function $\mathbb{R}\times\mathbb{R}\to\mathbb{R}_{\geq 0}$, by continuity, and so by Hilbert's 17th Problem, it's a sum of squares of rational functions (with real coefficients, but I'm willing to be that we can do it over $\mathbb{Q}$ as well).
