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

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 $m \leq2$, 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.