Real Algebraic and Analytic Geometry

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273. Claus Scheiderer:
Weighted sums of squares in local rings and their completions, I.


Submission: 2009, February 20.

Let A be an excellent local ring of real dimension at most 2, let T be a finitely generated preordering in A, and let T' be the preordering generated by T in the completion A' of A. Under a weak condition on the residue field k = A/m we show that T saturated implies T' saturated, and that a weak version of the converse holds as well. We also prove a transfer result between different real closed fields in the case where A is henselian and k is real closed. These results have direct implications for nonnegativity certificates for real polynomials which are nonnegative on suitable 2-dimensional semi-algebraic sets.

Mathematics Subject Classification (2000): 14P99,13H05,14P10,32S05.

Keywords and Phrases: Local rings, completion, Artin approximation, preorderings, curve singularities, positive polynomials, sums of squares, real algebraic geometry.

Full text, 20p.: dvi 173k, pdf 307k.

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