Quasiminimality for holomorphic functions

Dmitrieva, Anna (2025) Quasiminimality for holomorphic functions. Doctoral thesis, University of East Anglia.

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Abstract

In [Zil97] Zilber conjectured that the complex exponential field is quasiminimal, i.e., all definable subsets are countable or cocountable. This conjecture led to a more general conjecture that extending the complex field with any unary entire function, or even all of them simultaneously, makes it quasiminimal. We present two examples of additional analytical structure on the complex field which indeed turn it into a quasiminimal structure.

Our first example considers two elliptic curves and a correspondence between them. Then we show via the blurring method from [Kir19] that the complex field with the correspondence is quasiminimal under certain restrictions on the two elliptic curves. This result can be seen as a progress towards the Quasiminimality Conjecture for two Weierstrass functions.

For the second example we investigate the theory of a generic function as introduced by Zilber in [Zil02]. We provide a classification of types in this theory and deduce that the complex field with an entire generic function is quasiminimal. Furthermore, we show that any two such structures are isomorphic.

Additionally, we establish a general construction for a pregeometry given by a family of analytic functions on the real or the complex field. We give several characterizations of this pregeometry and conclude that whenever the family in question is countable, the corresponding pregeometry has the countable closure property. We provide an application of this construction by using these results to show quasiminimality of our first example.

Item Type:	Thesis (Doctoral)
Faculty \ School:	Faculty of Science > School of Engineering, Mathematics and Physics
Depositing User:	Chris White
Date Deposited:	06 May 2026 10:55
Last Modified:	06 May 2026 10:55
URI:	https://ueaeprints.uea.ac.uk/id/eprint/102911
DOI:

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