Polynomials in R[x,y] That Are Sums of Squares in R (x,y)

A positive semidefinite polynomial f ∈ R[x, y] is said to be ∑(m, n) if f is a sum of m squares in R(x, y), but no fewer, and f is a sum of n squares in R[x, y], but no fewer. If f is not a sum of polynomial squares, then we set n = ∞. It is known that if m ≤ 2, then m = n. The Motzkin polynomial is known to be Σ(4, ∞). We present a family of Σ(3, 4) polynomials and a family of Σ(3, ∞) polynomials. Thus, a positive semidefinite polynomial in 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.