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Israel Journal of Mathematics

, Volume 213, Issue 1, pp 139–173 | Cite as

Global complexification of real analytic globally subanalytic functions

  • Tobias KaiserEmail author
Article

Abstract

We show that a globally subanalytic function that is real analytic can be extended in a globally subanalytic way to a holomorphic function. We also establish a parametric version of this result. For other important ominimal structures, we show that unary definable real analytic functions can be extended definably to a holomorphic function.

Keywords

Holomorphic Function Real Analytic Function Cell Decomposition Function Germ Convergent Power Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    R. Bianconi, Nondefinability results for expansions of the field of real numbers by the exponential function and by the restricted sine function, Journal of Symbolic Logic 62 (1997), 1173–1178.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    R. Bianconi, Undefinability results in o-minimal expansions of the real numbers, Annals of Pure and Applied Logic 134 (2005), 43–51.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Institut des Hautes Études Scientifiques. Publications Mathématiques 62 (1988), 5–42.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    N. Bourbaki, General Topology. Chapters 1–4, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989.Google Scholar
  5. [5]
    J. Denef and L. van den Dries, p-adic and real subanalytic sets, Annals of Mathematics 128 (1988), 79–138.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. Denef and L. Lipshitz, Ultraproducts and approximation in local rings. II, Mathematische Annalen 253 (1980), 1–28.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    L. van den Dries, A generalization of the Tarski–Seidenberg theorem, and some nondefinability results, Bulletin of the American Mathematical Society 15 (1986), 189–193.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    L. van den Dries, On the elementary theory of restricted elementary functions, Journal of Symbolic Logic 53 (1988), 796–808.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    L. van den Dries, Tame Topology and O-minimal Structures, London Mathematical Society Lecture Notes Series, Vol. 248, Cambridge University Press, Cambridge, 1998.Google Scholar
  10. [10]
    L. van den Dries, A. Macintyre and D. Marker, The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics 140 (1994), 183–205.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    L. van den Dries and C. Miller, Extending Tamm’s theorem, Annales de l’Institut Fourier 44 (1994), 1367–1395.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    T. Kaiser, The Riemann mapping theorem for semianalytic domains and o-minimality, Proceedings of the London Mathematical Society 98 (2009), 427–444.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    T. Kaiser, Integration of semialgebraic functions and integrated Nash functions, Mathematische Zeitschrift 275 (2013), 349–366.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    T. Kaiser, Multivariate rings of Puiseux series induced by a Weierstrass system and twisted group rings, Communications in Algebra 42 (2014), 4619–4634.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S. Krantz and H. Parks, A Primer of Real Analytic Functions, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Boston, MA, 2002.Google Scholar
  16. [16]
    J.-M. Lion and J.-P. Rolin, Théorème de préparation pour les fonctions logarithmicoexponentielles, Annales de l’Institut Fourier 47 (1997), 859–884.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J.-M. Lion and J.-P. Rolin, Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Annales de l’Institut Fourier 48 (1998), 755–767.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    D. Miller, A preparation theorem for Weierstrass systems, Transactions of the American Mathematical Society 358 (2006), 4395–4439.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Y. Peterzil and S. Starchenko, Complex analytic geometry and analytic-geometric categories, Journal für die Reine und Angewandte Mathematik 626 (2009), 39–74.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Y. Peterzil and S. Starchenko, Tame complex analysis and o-minimality, in Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 58–81.Google Scholar
  21. [21]
    Y. Peterzil and S. Starchenko, Definability of restricted theta functions and families of abelian varieties, Duke Mathematical Journal 162 (2013), 731–765.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    M. Shiota, Geometry of Subanalytic and Semialgebraic Sets, Progress in Mathematics, Vol. 150, Birkhäuser Boston, Boston, MA, 1997.Google Scholar

Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Faculty of Computer Science and MathematicsUniversity of PassauPassauGermany

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