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Blurred complex exponentiation

  • Jonathan KirbyEmail author


It is shown that the complex field equipped with the approximate exponential map, defined up to ambiguity from a small group, is quasiminimal: every automorphism-invariant subset of \(\mathbb {C}\) is countable or co-countable. If the ambiguity is taken to be from a subfield analogous to a field of constants then the resulting blurred exponential field is isomorphic to the result of an equivalent blurring of Zilber’s exponential field, and to a suitable reduct of a differentially closed field. These results are progress towards Zilber’s conjecture that the complex exponential field itself is quasiminimal. A key ingredient in the proofs is to prove the analogue of the exponential-algebraic closedness property using the density of the group governing the ambiguity with respect to the complex topology.


Complex exponentiation Quasiminimal Ax-Schanuel Zilber conjecture 

Mathematics Subject Classification

Primary: 03C65 Secondary: 03C48 



I would like to thank Boris Zilber for many useful conversations. I learned the idea of blurring a model-theoretically wild structure to produce a stable structure from his paper [11]. I would also like to thank the many seminar audiences in the American Midwest who made helpful comments when I talked about these ideas in 2006/07.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsUniversity of East AngliaNorwichUK

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