Algebra & Number Theory

Pseudo-exponential maps, variants, and quasiminimality

Martin Bays and Jonathan Kirby

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Abstract

We give a construction of quasiminimal fields equipped with pseudo-analytic maps, generalizing Zilber’s pseudo-exponential function. In particular we construct pseudo-exponential maps of simple abelian varieties, including pseudo--functions for elliptic curves. We show that the complex field with the corresponding analytic function is isomorphic to the pseudo-analytic version if and only if the appropriate version of Schanuel’s conjecture is true and the corresponding version of the strong exponential-algebraic closedness property holds. Moreover, we relativize the construction to build a model over a fairly arbitrary countable subfield and deduce that the complex exponential field is quasiminimal if it is exponentially-algebraically closed. This property states only that the graph of exponentiation has nonempty intersection with certain algebraic varieties but does not require genericity of any point in the intersection. Furthermore, Schanuel’s conjecture is not required as a condition for quasiminimality.

Article information

Source
Algebra Number Theory, Volume 12, Number 3 (2018), 493-549.

Dates
Received: 17 March 2017
Revised: 13 November 2017
Accepted: 26 December 2017
First available in Project Euclid: 28 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1532743363

Digital Object Identifier
doi:10.2140/ant.2018.12.493

Mathematical Reviews number (MathSciNet)
MR3815305

Zentralblatt MATH identifier
06890760

Subjects
Primary: 03C65: Models of other mathematical theories
Secondary: 03C75: Other infinitary logic 12L12: Model theory [See also 03C60]

Keywords
exponential fields predimension categoricity Schanuel conjecture Ax–Schanuel Zilber–Pink quasiminimality Kummer theory

Citation

Bays, Martin; Kirby, Jonathan. Pseudo-exponential maps, variants, and quasiminimality. Algebra Number Theory 12 (2018), no. 3, 493--549. doi:10.2140/ant.2018.12.493. https://projecteuclid.org/euclid.ant/1532743363


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