Blurred complex exponentiation

• Jonathan Kirby ^[1]
  1. [1] University of East Anglia

     University of East Anglia

     Norwich District, Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 25, Nº. 5, 2019
• Idioma: inglés
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• Resumen

  • 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 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.