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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 \[ f+\varepsilon q\in \mathrm {PO}(f_1,\ldots ,f_s) \mbox { for all } \varepsilon >0.\] 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 \[ \mathrm {PO}(f_1,\ldots ,f_s)^{\vee \vee },\] 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

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.