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 \(\mathbb {C}\) 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.
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Notes
Thanks to Margaret Friedland who gave a helpful answer on MathOverflow to help fill a gap in an earlier version of this proof. https://mathoverflow.net/q/343181.
References
Ax, J.: On Schanuel’s conjectures. Ann. Math. 2(93), 252–268 (1971)
Ax, J.: Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups. Am. J. Math. 94, 1195–1204 (1972)
Bays, M., Kirby, J.: Pseudo-exponential maps, variants, and quasiminimality. Algebra Number Theory 12(3), 493–549 (2018)
Bombieri, E., Masser, D., Zannier, U.: Anomalous subvarieties—structure theorems and applications. Int. Math. Res. Not. IMRN (19):Art. ID rnm057, 33 (2007)
Kirby, J.: The theory of the exponential differential equations of semiabelian varieties. Sel. Math. NS 15(3), 445–486 (2009)
Kirby, J.: On quasiminimal excellent classes. J. Symb. Logic 75(2), 551–564 (2010)
Kirby, J., Zilber, B.: Exponentially closed fields and the conjecture on intersections with tori. Ann. Pure Appl. Logic 165(11), 1680–1706 (2014)
Łojasiewicz, Stanisław: Introduction to Complex Analytic Geometry. Birkhäuser Verlag, Basel (1991). Translated from the Polish by Maciej Klimek
Zilber, B.: Generalized analytic sets. Algebra i Logika 36(4), 387–406 (1997). 478
Zilber, B.: Fields with pseudo-exponentiation. arXiv:math/0012023 (2000)
Zilber, B.: Bi-coloured fields on the complex numbers. J. Symb. Logic 69(4), 1171–1186 (2004)
Zilber, B.: Pseudo-exponentiation on algebraically closed fields of characteristic zero. Ann. Pure Appl. Logic 132(1), 67–95 (2005)
Acknowledgements
I would like to thank Boris Zilber for many useful conversations. I learned the idea of blurring a model-theoretically wild structure to produce a stable structure from his paper [11]. I would also like to thank the many seminar audiences in the American Midwest who made helpful comments when I talked about these ideas in 2006/07.
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Kirby, J. Blurred complex exponentiation. Sel. Math. New Ser. 25, 72 (2019). https://doi.org/10.1007/s00029-019-0517-4
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DOI: https://doi.org/10.1007/s00029-019-0517-4