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  • Sums of squares and moment problems in equivariant situations
    Cimprič, Jaka ; Kuhlmann, Salma, 1958- ; Scheiderer, Claus, 1959-
    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.
    Source: Transactions of the American Mathematical Society. - ISSN 0002-9947 (Vol. 361, no. 2, 2009, str. 735-765)
    Type of material - article, component part
    Publish date - 2009
    Language - english
    COBISS.SI-ID - 15111769

    Link(s):

    http://dx.doi.org/10.1090/S0002-9947-08-04588-1

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