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Positivity

, Volume 15, Issue 2, pp 297–307 | Cite as

Positivstellensätze for differentiable functions

  • Andreas Fischer
Article

Abstract

We present a canonical proof of both the strict and weak Positivstellensatz for rings of differentiable and smooth functions. Our construction is explicit, preserves definability in expansions of the real field, and it works in definably complete expansions of real closed fields as well as for real-valued functions on Banach spaces.

Keywords

Positivstellensatz Differentiable and smooth function Definably complete structure Banach space 

Mathematics Subject Classification (2000)

Primary 03C64 14P10 Secondary 46E25 26E40 

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References

  1. 1.
    Acquistapace F., Andradas C., Broglia F.: The Positivstellensatz for definable functions on o-minimal structures. Ill. J. Math. 46(3), 685–693 (2002)MathSciNetMATHGoogle Scholar
  2. 2.
    Bochnak, J., Coste, M., Roy, M.-F.: Real algebraic geometry. Translated from the 1987 French original. Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36. Springer, Berlin, x+430 pp (1998)Google Scholar
  3. 3.
    Coste, M.: An Introduction to O-minimal Geometry. Dip. Mat. Univ. Pisa. Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa (2000)Google Scholar
  4. 4.
    van den Dries, L.: Tame topology and O-minimal structures. LMS Lecture Notes, vol. 248. Cambridge University Press (1998)Google Scholar
  5. 5.
    Krivine J.L.: Anneau preordones. J. Anal. Math. 12, 307–326 (1964)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Miller C.: Expansions of dense linear orders with the intermediate value property. J. Symb. Logic 66(4), 1783–1790 (2001)MATHCrossRefGoogle Scholar
  7. 7.
    Stengle G.: A nullstellensatz and a positivstellensatz in semialgebraic geometry. Math. Ann. 207, 87–97 (1974)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Schmüdgen K.: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289(2), 203–206 (1991)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Fields InstituteTorontoCanada
  2. 2.Gymnasium St. Ursula DorstenDorstenGermany

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