# Model Theory of Holomorphic Functions

Braun, H. T. F. (2004) Model Theory of Holomorphic Functions. PhD thesis, University of Oxford.

 Preview

650kB

## Abstract

This thesis is concerned with a conjecture of Zilber: that the complex field expanded with the exponential function should be quasi-minimal'; that is, all its definable subsets should be countable or have countable complement. Our purpose is to study the geometry of this structure and other expansions by holomorphic functions of the complex field without having first to settle any number-theoretic problems, by treating all countable sets on an equal footing.

We present axioms, modelled on those for a Zariski geometry, defining a non-first-order class of `quasi-Zariski'' structures endowed with a dimension theory and a topology in which all countable sets are of dimension zero. We derive a quantifier elimination theorem, implying that members of the class are quasi-minimal.

We look for analytic structures in this class. To an expansion of the complex field by entire holomorphic functions we associate a sheaf of analytic germs which is closed under application of the implicit function theorem. We prove that is also closed under partial differentiation and that it admits Weierstrass preparation. The sheaf defines a subclass of the analytic sets which we call -analytic. We develop analytic geometry for this class proving a Nullstellensatz and other classical properties. We isolate a condition on the asymptotes of the varieties of certain functions in . If this condition is satisfied then the -analytic sets induce a quasi-Zariski structure under countable union. In the motivating case of the complex exponential we prove a low-dimensional case of the condition, towards the original conjecture.

Item Type: Thesis (PhD) H - N > Mathematical logic and foundationsD - G > Functions of a complex variableO - Z > Several complex variables and analytic spaces Mathematical Logic Group 105 Eprints Administrator 19 Jul 2004 29 May 2015 18:16

Repository Staff Only: item control page