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 language | English |
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Article number | 72 |
Journal | Selecta Mathematica-New Series |
Volume | 25 |
Issue number | 5 |
Early online date | 13 Nov 2019 |
DOIs | |
Publication status | Published - Dec 2019 |
Keywords
- Ax-Schanuel
- Complex exponentiation
- Quasiminimal
- Zilber conjecture
Profiles
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Jonathan Kirby
- School of Engineering, Mathematics and Physics - Reader
- Logic - Member
Person: Research Group Member, Academic, Teaching & Research