TITLE: Are all polynomial inequalities deducible from the trivial inequality?
QUESTION [37 upvotes]: I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of polynomials in the variables $x_i$.  Unfortunately, I'm not sure if I'm remembering correctly.  (The context in which I saw this theorem was someone asking whether there was a sum-of-squares proof of the AM-GM inequality in $n$ variables, so I'm not 100% certain if the quoted theorem was specific to that case.)
So: does anyone know a reference for the correct statement of this theorem, if in fact something like it is true?  (Feel free to retag if you don't think it's appropriate, by the way.)

REPLY [19 votes]: Let me draw your attention to the paper "There are significantly more nonegative polynomials than sums of squares" by Grigoriy Blekherman. Here are slides of a related talk by Grigoriy.
The abstract reads: "We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of even powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinity. When the degree is larger than two, it follows that there are significantly more nonnegative polynomials than sums of squares and there are significantly more sums of squares than sums of even powers of linear forms. Moreover, we quantify the exact discrepancy between the cones; from our bounds it follows that the discrepancy grows as the number of variables increases."