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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Positive polynomials and sequential closures of quadratic modules
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by Tim Netzer PDF
Trans. Amer. Math. Soc. 362 (2010), 2619-2639 Request permission

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
  • Email: tim.netzer@math.uni-leipzig.de
  • Received by editor(s): July 21, 2008
  • Published electronically: December 14, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 2619-2639
  • MSC (2000): Primary 44A60, 14P10, 13J30; Secondary 11E25
  • DOI: https://doi.org/10.1090/S0002-9947-09-05001-6
  • MathSciNet review: 2584613