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.