Notre Dame Journal of Formal Logic

A Note on the Axioms for Zilber’s Pseudo-Exponential Fields

Jonathan Kirby


We show that Zilber’s conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in pseudo-exponentiation leads to a description of the elementary embeddings, and the result that pseudo-exponential fields are precisely the models of their common first-order theory which are atomic over exponential transcendence bases. We also show that the class of all pseudo-exponential fields is an example of a nonfinitary abstract elementary class, answering a question of Kesälä and Baldwin.

Article information

Notre Dame J. Formal Logic, Volume 54, Number 3-4 (2013), 509-520.

First available in Project Euclid: 9 August 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C65: Models of other mathematical theories
Secondary: 03C48: Abstract elementary classes and related topics [See also 03C45]

pseudo-exponentiation exponential fields Schanuel property first-order theory abstract elementary class


Kirby, Jonathan. A Note on the Axioms for Zilber’s Pseudo-Exponential Fields. Notre Dame J. Formal Logic 54 (2013), no. 3-4, 509--520. doi:10.1215/00294527-2143844.

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  • [1] Hyttinen, T., and M. Kesälä, “Independence in finitary abstract elementary classes,” Annals of Pure and Applied Logic, vol. 143 (2006), pp. 103–38.
  • [2] Kirby, J., “Exponential algebraicity in exponential fields,” Bulletin of the London Mathematical Society, vol. 42 (2010), pp. 879–90.
  • [3] Kirby, J., “Finitely presented exponential fields,” to appear in Algebra and Number Theory, preprint, arXiv:0912.4019v4 [math.LO].
  • [4] Kirby, J., and B. Zilber, “Exponential fields and atypical intersections,” preprint, arXiv:1108.1075v1 [math.LO].
  • [5] Kueker, D. W., “Abstract elementary classes and infinitary logics,” Annals of Pure and Applied Logic, vol. 156 (2008), pp. 274–86.
  • [6] Macintyre, A. J., “Exponential algebra,” pp. 191–210 in Logic and Algebra (Pontignano, 1994), vol. 180 of Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1996.
  • [7] Zilber, B., “Pseudo-exponentiation on algebraically closed fields of characteristic zero,” Annals of Pure and Applied Logic, vol. 132 (2005), pp. 67–95.
  • [8] Zilber, B., “Covers of the multiplicative group of an algebraically closed field of characteristic zero,” Journal of the London Mathematical Society (2), vol. 74 (2006), pp. 41–58.