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Lectures on the Model Theory of Real and Complex Exponentiation

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Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 2111)

Abstract

In these notes we sketch a proof of the model completeness of the real exponential field. We begin with an introduction to the various Preparation Theorems required for the proof as well as a discussion of polynomially bounded, o-minimal structures. We then discuss the appropriate valuation theoretic setting and show how the so-called Valuation Inequality leads to the desired result. We conclude with some speculative remarks on the model theory of the complex exponential field.

Keywords

  • Complex Exponentiation
  • Real Exponential field
  • Definable Closure
  • Khovanski
  • Quasipolynomial Map

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. J. Denef, L. van den Dries, p-Adic and real subanalytic sets. Ann. Math. 128, 79–138 (1988)

    Google Scholar 

  2. A. Gabrielov, Projections of semi-analytic sets. Funct. Anal. Appl. 2, 282–291 (1968)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. A. Gabrielov, Complements of subanalytic sets and existential formulas for analytic functions. Inv. Math. 125, 1–12 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. G.O. Jones, A.J. Wilkie, Locally polynomially bounded structures. Bull. Lond. Math. Soc. 40(2), 239–248 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. A. Khovanski, On a class of systems of transcendental equations. Sov. Math. Dokl. 22, 762–765 (1980)

    Google Scholar 

  6. J.F. Knight, A. Pillay, C. Steinhorn, Definable sets in ordered structures, II. Trans. Am. Math. Soc. 295, 593–605 (1986)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. D. Marker, A remark on Zilber’s pseudoexponentiation. J Symb. Log. 71(3), 791–798 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. A. Pillay, C. Steinhorn, Definable sets in ordered structures I. Trans. Amer. Math. Soc. 295, 565–592 (1986)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Y. Peterzil, S. Starchenko, Expansions of algebraically closed fields in o-minimal structures. Sel. Math. New Ser. 7, 409–445 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. J.M. Ruiz, The Basic Theory of Power Series (Edizioni ETS, Pisa, 2009)

    Google Scholar 

  11. P. Speissegger, in Lectures on o-minimality, ed. by B. Hart, M. Valeriote. Lectures on Algebraic Model Theory, Fields Institute Monographs (American Mathematical Society, Providence, 2000)

    Google Scholar 

  12. L. van den Dries, On the elementary theory of restricted elementary functions. J. Symb. Log. 53, 796–808 (1988)

    CrossRef  MATH  Google Scholar 

  13. L. van den Dries, C. Miller, On the real exponential field with restricted analytic functions. Isr. J. Math 85, 19–56 (1994)

    CrossRef  MATH  Google Scholar 

  14. L. van den Dries, Tame Topology and o-minimal Structures. London Mathematical Society Lecture Note Series, vol. 248 (Cambridge University Press, Cambridge, 1998)

    Google Scholar 

  15. A.J. Wilkie, On the theory of the real exponential field. Ill. J. Math. 33(3), 384–408 (1989)

    MathSciNet  MATH  Google Scholar 

  16. A.J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Am. Math. Soc. 9(4), 1051–1094 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. A.J. Wilkie, Model theory of analytic and smooth functions, in Models and Computability, ed. by S.B. Cooper, J.K. Truss. LMS Lecture Note Series, vol. 259, pp. 407–419 (1999)

    Google Scholar 

  18. A.J. Wilkie, Some results and problems on complex germs with definable Mittag-Leffler stars. Notre Dame J. Formal Logic 54(3-4), 603–610 (2013)

    Google Scholar 

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Wilkie, A.J. (2014). Lectures on the Model Theory of Real and Complex Exponentiation. In: Model Theory in Algebra, Analysis and Arithmetic. Lecture Notes in Mathematics(), vol 2111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54936-6_3

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