Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11E25, 12D15

Retrieve articles in all journals with MSC (2000): 11E25, 12D15

Additional Information

David B. Leep
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

Colin L. Starr
Affiliation: Department of Mathematics and Statistics, Box 13040 SFA Station, Stephen F. Austin State University, Nacogdoches, Texas 75962-3040

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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia