## 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.## References

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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