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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Polynomials in $\mathbb {R}[x,y]$ that are sums of squares in $\mathbb {R}(x,y)$
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by David B. Leep and Colin L. Starr PDF
Proc. Amer. Math. Soc. 129 (2001), 3133-3141 Request permission

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.
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Additional Information
  • David B. Leep
  • Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
  • Email: leep@ms.uky.edu
  • Colin L. Starr
  • Affiliation: Department of Mathematics and Statistics, Box 13040 SFA Station, Stephen F. Austin State University, Nacogdoches, Texas 75962-3040
  • Email: starr@math.sfasu.edu
  • Received by editor(s): May 19, 1999
  • Received by editor(s) in revised form: March 8, 2000
  • Published electronically: April 9, 2001
  • Additional Notes: This work formed part of the second author’s dissertation.
  • Communicated by: Lance W. Small
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 3133-3141
  • MSC (2000): Primary 11E25, 12D15
  • DOI: https://doi.org/10.1090/S0002-9939-01-05927-5
  • MathSciNet review: 1844985