nt.number theory - 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.