
> 
> 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.



Nonnegative polynomial in two variables
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. 


