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Transactions of the American Mathematical Society

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Sums of squares and moment problems in equivariant situations

Authors: Jaka Cimpric, Salma Kuhlmann and Claus Scheiderer
Journal: Trans. Amer. Math. Soc. 361 (2009), 735-765
MSC (2000): Primary 14P10, 14L30, 20G20
Published electronically: September 23, 2008
MathSciNet review: 2452823
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Abstract: We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group $ G$ over $ \mathbb{R}$ acting on an affine $ \mathbb{R}$-variety $ V$, we consider the induced dual action on the coordinate ring $ \mathbb{R}[V]$ and on the linear dual space of $ \mathbb{R}[V]$. In this setting, given an invariant closed semialgebraic subset $ K$ of $ V(\mathbb{R})$, we study the problem of representation of invariant nonnegative polynomials on $ K$ by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on $ \mathbb{R}[V]$ by invariant measures supported on $ K$. To this end, we analyse the relation between quadratic modules of $ \mathbb{R}[V]$ and associated quadratic modules of the (finitely generated) subring $ \mathbb{R}[V]^G$ of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional $ K$-moment problem. Most of our results are specific to the case where the group $ G(\mathbb{R})$ is compact.

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Additional Information

Jaka Cimpric
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenija

Salma Kuhlmann
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Room 142 McLean Hall, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6

Claus Scheiderer
Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, D-78457 Konstanz, Germany

Keywords: Semi-algebraic sets, group actions, moment problems
Received by editor(s): November 19, 2006
Published electronically: September 23, 2008
Additional Notes: The third author was partially supported by the European RTNetwork RAAG, HPRN-CT-2001-00271
Article copyright: © Copyright 2008 American Mathematical Society

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