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

Keywords

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

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