Polynomials in $\mathbb {R}[x,y]$ that are sums of squares in $\mathbb {R}(x,y)$

Abstract:

A positive semidefinite polynomial $f \in \mathbb {R}[x,y]$ is said to be $\Sigma (m,n)$ if $f$ is a sum of $m$ squares in $\mathbb {R}(x,y)$, but no fewer, and $f$ is a sum of $n$ squares in $\mathbb {R}[x,y]$, but no fewer. If $f$ is not a sum of polynomial squares, then we set $n=\infty$. It is known that if $m \leq 2$, then $m=n$. The Motzkin polynomial is known to be $\Sigma (4,\infty )$. We present a family of $\Sigma (3,4)$ polynomials and a family of $\Sigma (3, \infty )$ polynomials. Thus, a positive semidefinite polynomial in $\mathbb {R}[x,y]$ may be a sum of three rational squares, but not a sum of polynomial squares. This resolves a problem posed by Choi, Lam, Reznick, and Rosenberg.