An elementary and constructive solution to Hilbert’s 17th Problem for matrices
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- by Christopher J. Hillar and Jiawang Nie PDF
- Proc. Amer. Math. Soc. 136 (2008), 73-76 Request permission
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.References
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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
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 73-76
- MSC (2000): Primary 12D15, 03C64, 13L05, 14P05, 15A21, 15A54
- DOI: https://doi.org/10.1090/S0002-9939-07-09068-5
- MathSciNet review: 2350390