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Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets

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Abstract

We prove a criterion for an element of a commutative ring to be contained in an archimedean subsemiring. It can be used to investigate the question whether nonnegativity of a polynomial on a compact semialgebraic set can be certified in a certain way. In case of (strict) positivity instead of nonnegativity, our criterion simplifies to classical results of Stone, Kadison, Krivine, Handelman, Schmüdgen et al. As an application of our result, we give a new proof of the following result of Handelman: If an odd power of a real polynomial in several variables has only nonnegative coefficients, then so do all sufficiently high powers.

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Authors and Affiliations

  1. Fachbereich Mathematik und Statistik, Universität Konstanz, 78457, Konstanz, Germany

    Markus Schweighofer

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  1. Markus Schweighofer
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Correspondence to Markus Schweighofer.

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Partially supported by the DFG project 214371 “Darstellung positiver Polynome”.

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Schweighofer, M. Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets. manuscripta math. 117, 407–428 (2005). https://doi.org/10.1007/s00229-005-0568-z

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  • DOI: https://doi.org/10.1007/s00229-005-0568-z

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