manuscripta mathematica

, Volume 119, Issue 4, pp 395–410 | Cite as

Sums of squares on real algebraic surfaces



Consider real polynomials g1, . . . , gr in n variables, and assume that the subset K = {g1≥0, . . . , gr≥0} of ℝn is compact. We show that a polynomial f has a representation

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in which the se 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.

Mathematics Subject Classification (2000)

Primary 14P05 secondary 11E25 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Fachbereich Mathematik und StatistikUniversität KonstanzKonstanzGermany

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