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Weighted sums of squares in local rings and their completions, I

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Abstract

Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let \({\widehat{T}}\) be the preordering generated by T in the completion \({\widehat{A}}\) . Under a weak condition on the residue field \({k=A/\mathfrak{m}}\) we show that T saturated implies \({\widehat{T}}\) 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 two-dimensional semi-algebraic sets.

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Authors and Affiliations

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

    Claus Scheiderer

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  1. Claus Scheiderer
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Correspondence to Claus Scheiderer.

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The main results were obtained when the European RTN network HPRN-CT-2001-00271 (Real Algebraic and Analytic Geometry) was still running. Support by this network is gratefully acknowledged. Part of this work was carried out when the author enjoyed a stay at MSRI Berkeley in 2004. He would like to thank the institute for hospitality and excellent working conditions.

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Scheiderer, C. Weighted sums of squares in local rings and their completions, I. Math. Z. 266, 1–19 (2010). https://doi.org/10.1007/s00209-009-0551-6

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  • DOI: https://doi.org/10.1007/s00209-009-0551-6

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