Polynomials in ℝ[x,y] that are sums of squares in ℝ(x,y)

David B. Leep, Colin L. Starr

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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 ∑(4,∞). We present a family of 2(3,4) polynomials and a family of ∑(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.

Original languageEnglish
Pages (from-to)3133-3141
Number of pages9
JournalProceedings of the American Mathematical Society
Issue number11
StatePublished - 2001


  • Positive semidcfinite polynomial
  • Sum of squares of polynomials

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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