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Blurred Complex Exponentiation

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Blurred Complex Exponentiation. / Kirby, Jonathan.

In: Selecta Mathematica, Vol. 25, No. 5, 72, 12.2019.

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@article{77b10bc5b5db4456a6aa1f2df9b719b4,
title = "Blurred Complex Exponentiation",
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.",
keywords = "Ax-Schanuel, Complex exponentiation, Quasiminimal, Zilber conjecture",
author = "Jonathan Kirby",
year = "2019",
month = dec,
doi = "10.1007/s00029-019-0517-4",
language = "English",
volume = "25",
journal = "Selecta Mathematica",
issn = "1022-1824",
publisher = "Birkhauser Verlag Basel",
number = "5",

}

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TY - JOUR

T1 - Blurred Complex Exponentiation

AU - Kirby, Jonathan

PY - 2019/12

Y1 - 2019/12

N2 - 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.

AB - 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.

KW - Ax-Schanuel

KW - Complex exponentiation

KW - Quasiminimal

KW - Zilber conjecture

UR - http://www.scopus.com/inward/record.url?scp=85075160424&partnerID=8YFLogxK

U2 - 10.1007/s00029-019-0517-4

DO - 10.1007/s00029-019-0517-4

M3 - Article

VL - 25

JO - Selecta Mathematica

JF - Selecta Mathematica

SN - 1022-1824

IS - 5

M1 - 72

ER -

ID: 129743642