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

 

An elementary and constructive solution to Hilbert's 17th Problem for matrices


Authors: Christopher J. Hillar and Jiawang Nie
Journal: Proc. Amer. Math. Soc. 136 (2008), 73-76
MSC (2000): Primary 12D15, 03C64, 13L05, 14P05, 15A21, 15A54
Published electronically: October 12, 2007
MathSciNet review: 2350390
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Abstract: We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let $ A$ be an $ n \times n$ symmetric matrix with entries in the polynomial ring $ \mathbb{R}[x_1,\ldots,x_m]$. The result is that if $ A$ is positive semidefinite for all substitutions $ (x_1,\ldots,x_m) \in \mathbb{R}^m$, then $ A$ can be expressed as a sum of squares of symmetric matrices with entries in $ \mathbb{R}(x_1,\ldots,x_m)$. Moreover, our proof is constructive and gives explicit representations modulo the scalar case.


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Additional Information

Christopher J. Hillar
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: chillar@math.tamu.edu

Jiawang Nie
Affiliation: Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455
Email: njw@ima.umn.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09068-5
PII: S 0002-9939(07)09068-5
Keywords: Artin's theorem, Hilbert's 17th problem, sums of squares, positive semidefinite matrix, real closed field
Received by editor(s): October 23, 2006
Received by editor(s) in revised form: December 14, 2006
Published electronically: October 12, 2007
Additional Notes: The first author is supported under an NSF Postdoctoral Research Fellowship. This research was conducted during the Positive Polynomials and Optimization workshop at the Banff International Research Station, October 7–12 (2006), Banff, Canada.
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.