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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Positive polynomials and sequential closures of quadratic modules


Author: Tim Netzer
Journal: Trans. Amer. Math. Soc. 362 (2010), 2619-2639
MSC (2000): Primary 44A60, 14P10, 13J30; Secondary 11E25
Published electronically: December 14, 2009
MathSciNet review: 2584613
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Abstract: Let $ \mathcal{S}=\{x\in\mathbb{R}^n\mid f_1(x)\geq 0,\ldots,f_s(x)\geq 0\}$ be a basic closed semi-algebraic set in $ \mathbb{R}^n$ and let $ \mathrm{PO}(f_1,\ldots,f_s)$ be the corresponding preordering in $ \mathbb{R}[X_1,\ldots,X_n]$. We examine for which polynomials $ f$ there exist identities

$\displaystyle f+\varepsilon q\in \mathrm{PO}(f_1,\ldots,f_s)$    for all $\displaystyle \varepsilon>0.$

These are precisely the elements of the sequential closure of $ \mathrm{PO}(f_1,\ldots,f_s)$ with respect to the finest locally convex topology. We solve the open problem from Kuhlmann, Marshall, and Schwartz (2002, 2005), whether this equals the double dual cone

$\displaystyle \mathrm{PO}(f_1,\ldots,f_s)^{\vee\vee},$

by providing a counterexample. We then prove a theorem that allows us to obtain identities for polynomials as above, by looking at a family of fibre-preorderings, constructed from bounded polynomials. These fibre-preorderings are easier to deal with than the original preordering in general. For a large class of examples we are thus able to show that either every polynomial $ f$ that is nonnegative on $ \mathcal{S}$ admits such representations, or at least the polynomials from $ \mathrm{PO}(f_1,\ldots,f_s)^{\vee\vee}$ do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings.


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Additional Information

Tim Netzer
Affiliation: Fakultät für Mathematik und Informatik, Universität Leipzig, PF 100920, 04009 Leipzig, Germany
Email: tim.netzer@math.uni-leipzig.de

DOI: http://dx.doi.org/10.1090/S0002-9947-09-05001-6
PII: S 0002-9947(09)05001-6
Keywords: Moment problems, semi-algebraic sets, real algebra, positive polynomials and sum of squares
Received by editor(s): July 21, 2008
Published electronically: December 14, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.