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

DOI:
https://doi.org/10.1090/S0002-9947-09-05001-6

Published electronically:
December 14, 2009

MathSciNet review:
2584613

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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.