Positive polynomials and sequential closures of quadratic modules
Author:
Tim Netzer
Journal:
Trans. Amer. Math. Soc. 362 (2010), 26192639
MSC (2000):
Primary 44A60, 14P10, 13J30; Secondary 11E25
Published electronically:
December 14, 2009
MathSciNet review:
2584613
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Abstract: Let be a basic closed semialgebraic set in and let be the corresponding preordering in . We examine for which polynomials there exist identities for all These are precisely the elements of the sequential closure of 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 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 fibrepreorderings, constructed from bounded polynomials. These fibrepreorderings 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 that is nonnegative on admits such representations, or at least the polynomials from 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.unileipzig.de
DOI:
http://dx.doi.org/10.1090/S0002994709050016
PII:
S 00029947(09)050016
Keywords:
Moment problems,
semialgebraic 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.
