# Blurred Complex Exponentiation

## TL;DR

This paper investigates a modified complex exponential structure with an ambiguity component, demonstrating its quasiminimality and connections to Zilber's exponential field, advancing the understanding of complex exponential algebraic properties.

## Contribution

It introduces the concept of a blurred exponential field with an ambiguity group, proving its quasiminimality and relating it to Zilber's exponential field and differentially closed fields.

## Key findings

- The blurred exponential field is quasiminimal.
- It is isomorphic to a blurring of Zilber's exponential field.
- The approach uses density of the ambiguity group in the complex topology.

## 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 the field 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.