>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 $\mathbb Z\times \mathbb Z\to\mathbb N$](https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers), 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.