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Blurred Complex Exponentiation

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Abstract

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 the field 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.

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Original languageEnglish
Article number72
JournalSelecta Mathematica
Volume25
Issue number5
Early online date13 Nov 2019
DOIs
Publication statusPublished - Dec 2019
Peer-reviewedYes

Keywords

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  • Ax-Schanuel, Complex exponentiation, Quasiminimal, Zilber conjecture

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