Analytic continuation and Zilber's quasiminimality conjecture

TL;DR

The paper proposes a strategy to prove Zilber's quasiminimality conjecture for the complex exponential field by leveraging analytic continuation properties within o-minimal structures, applicable to expansions by countable sets of entire functions.

Contribution

It introduces a novel approach connecting analytic continuation in o-minimal structures to the quasiminimality conjecture for complex exponential fields.

Findings

Strategy applies to expansions by countable sets of entire functions

Connects analytic continuation properties with model-theoretic minimality

Discusses current limitations in unconditional results

Abstract

In this article, which is dedicated to my friend and colleague Boris Zilber on the occasion of his 75th birthday, I put forward a strategy for proving his quasiminimality conjecture for the complex exponential field. That is, for showing that every subset of $\mathbb{C}$ definable in the expansion of the complex field by the complex exponential function is either countable or cocountable. In fact the strategy applies to any expansion of the complex field by a countable set of entire functions (in any number of variables) and is based on a certain property-an analytic continuation property-of the o-minimal structure obtained by expanding the ordered field of real numbers by the restrictions to compact boxes of the real and imaginary parts of the functions in the given set. In a final section I discuss briefly the (rather limited) extent of our unconditional knowledge in the area.