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Sums of squares on real algebraic surfaces

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Abstract

Consider real polynomials g 1, . . . , g r in n variables, and assume that the subset K = {g 1≥0, . . . , g r ≥0} of ℝn is compact. We show that a polynomial f has a representation

in which the s e are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal. We apply this result to provide large and concrete families of cases in which dim (K) = 2 and every polynomial f with f| K ≥0 has a representation (*). Before, it was not known whether a single such example exists. Further geometric and arithmetic applications are given.

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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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Support by DFG travel grant KON 1823/2002 and by the European RAAG network HPRN-CT-2001-00271 is gratefully acknowledged. Part of this work was done while the author enjoyed a stay at MSRI Berkeley. He would like to thank the institute for the invitation and the very pleasant working conditions.

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Scheiderer, C. Sums of squares on real algebraic surfaces. manuscripta math. 119, 395–410 (2006). https://doi.org/10.1007/s00229-006-0630-5

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  • DOI: https://doi.org/10.1007/s00229-006-0630-5

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