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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: David B. Leep and Colin L. Starr
Journal: Proc. Amer. Math. Soc. 129 (2001), 3133-3141
MSC (2000): Primary 11E25, 12D15
Published electronically: April 9, 2001
MathSciNet review: 1844985
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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 $m \leq2$, 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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05927-5
PII: S 0002-9939(01)05927-5
Keywords: Positive semidefinite polynomial, sum of squares of polynomials
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
Article copyright: © Copyright 2001 American Mathematical Society