A special case of quasiminimality

TL;DR

This paper proves a specific case of Zilber's quasiminimality conjecture for the complex exponential field, focusing on subsets defined by existential formulas involving polynomial and exponential terms.

Contribution

It establishes a particular instance of quasiminimality for the complex exponential field using prior work by Henson and Rubel.

Findings

Proves a special case of Zilber's quasiminimality conjecture.

Focuses on subsets defined by existential formulas with polynomial and exponential components.

Utilizes existing results to advance understanding of the complex exponential field.

Abstract

We deduce a special case of Zilber's quasiminimality conjecture, for the complex exponential field, from work of Henson and Rubel. Specifically, we deal with those subsets of $\mathbb{C}$ defined by formulas of the form $\exists\bar{y}(P(x,\bar{y})=0)$, where $P$ is a term formed from the language $\{+,\times,\exp\}$ together with parameters from $\mathbb{C}$.