Real Algebraic and Analytic Geometry |

Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets.

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homepage: http://perso.univ-rennes1.fr/markus.schweighofer/

Submission: 2005, April 17.

*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.*

Mathematics Subject Classification (2000): 13J25, 13J30, 16Y60, 26C99, 54H10.

Keywords and Phrases: nonnegative polynomial, semiring, preprime, preorder, preordering, archimedean, Real Representation Theorem, Kadison-Dubois Theorem.

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