Polynomials in \(\mathbb{R}[x,y]\) that are sums of squares in \(\mathbb{R}(x,y)\) (Q2731900)

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 \(n = \infty\). NEWLINENEWLINENEWLINEWe 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.