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 | References | Similar Articles | Additional Information

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 be an symmetric matrix with entries in the polynomial ring . The result is that if is positive semidefinite for all substitutions , then can be expressed as a sum of squares of symmetric matrices with entries in . 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

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.