Some Positivstellensätze in real closed valued fields

Details Some Positivstellensätze in real closed valued fields Some Positivstellensätze in real closed valued fields

2011-01-31

arxiv.org/abs/1101.5872v3

Mathematics

Algebraic Geometry

Scientific paper

The purpose of this paper is to give a generalization of Hilbert's seventeenth problem in real closed valued fields. That is, for some definable set of the form $\{x \in K^n | \nu(\vec{p}(x))\ge \nu(\vec{q}(x)) \} $, we would like to give an algebraic characterization of the set of polynomials which get only non-negative values on it. Using model theoretic methods, we give a general characterization of the positive semi-definite polynomials for any definable set with a Ganzstellensatz, and we also give a representation of those polynomials in the sense of Hilbert 17th problem (that is, in terms of sums of squares) for definable sets from a certain kind.

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