Israel Journal of Mathematics

, Volume 85, Issue 1–3, pp 19–56 | Cite as

On the real exponential field with restricted analytic functions

  • Lou van den Dries
  • Chris Miller


The model-theoretic structure (ℝan, exp) is investigated as a special case of an expansion of the field of reals by certain families ofC -functions. In particular, we use methods of Wilkie to show that (ℝan, exp) is (finitely) model complete and O-minimal. We also prove analytic cell decomposition and the fact that every definable unary function is ultimately bounded by an iterated exponential function.


Regular Solution Model Completeness Noetherian Ring Cell Decomposition Divisible Group 
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  1. [D-vdD] J. Denef and L. van den Dries,P-adic and real subanalytic sets, Ann. Math.128 (1988), 79–138.CrossRefGoogle Scholar
  2. [vdD1] L. van den Dries,Algebraic theories with definable Skolem functions, Journal of Symbolic Logic49 (1984), 625–629.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [vdD2] L. van den Dries,A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. AMS15 (1986), 189–193.zbMATHGoogle Scholar
  4. [vdD3] L. van den Dries,The elementary theory of restricted elementary functions, J. Symb. Logic53 (1988), 796–808.zbMATHCrossRefGoogle Scholar
  5. [vdD-M-M] L. van den Dries, A. Macintyre and D. Marker,The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics, to appear.Google Scholar
  6. [F] J. Frisch,Points de platitude d'un morphisme d'espaces analytiques, Inv. Math.4 (1967), 118–138.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [H] A. G. Hovanskii,On a class of systems of transcendental equations, Soviet Math. Dokl.22 (1980), 762–765.Google Scholar
  8. [K-P-S] J. Knight, A. Pillay and C. Steinhorn,Definable sets in ordered structures. II, Trans. AMS295 (1986), 593–605.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [P-S] A. Pillay and C. Steinhorn,Definable sets in ordered structures. I, Trans. AMS295 (1986), 565–592.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Re] J.-P. Ressayre,Integer parts of real closed exponential fields, preprint.Google Scholar
  11. [Ro] M. Rosenlicht,The rank of a Hardy field, Trans. AMS280 (1983), 659–671.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [W1] A. J. Wilkie,Model completeness results for expansions of the real ordered field I: Restricted Pfaffian functions, preprint 1991.Google Scholar
  13. [W2] A. J. Wilkie,Model completeness results for expansions of the real field II: the exponential function, preprint 1992.Google Scholar

Copyright information

© Hebrew University 1994

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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