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

 

On the absence of uniform denominators in Hilbert's 17th problem


Author: Bruce Reznick
Journal: Proc. Amer. Math. Soc. 133 (2005), 2829-2834
MSC (2000): Primary 11E10, 11E25, 11E76, 12D15, 14P99
Published electronically: March 24, 2005
MathSciNet review: 2159759
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Abstract: Hilbert showed that for most $(n,m)$ there exist positive semidefinite forms $p(x_1,\dots,x_n)$ of degree $m$ which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form $h$ so that $h^2p$ is a sum of squares of forms; that is, $p$ is a sum of squares of rational functions with denominator $h$. We show that, for every such $(n,m)$ there does not exist a single form $h$ which serves in this way as a denominator for every positive semidefinite $p(x_1,\dots,x_n)$ of degree $m$.


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Additional Information

Bruce Reznick
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: reznick@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07879-2
PII: S 0002-9939(05)07879-2
Received by editor(s): May 19, 2003
Received by editor(s) in revised form: May 24, 2004
Published electronically: March 24, 2005
Additional Notes: This material is based in part upon work of the author, supported by the USAF under DARPA/AFOSR MURI Award F49620-02-1-0325. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of these agencies.
Communicated by: Michael Stillman
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.