Exponential algebraicity in exponential fields

TL;DR

This paper provides an algebraic proof that the exponential algebraic closure in exponential fields forms a pregeometry and explores its dimension function, leading to implications for Schanuel's conjecture.

Contribution

It introduces an algebraic proof of the pregeometry property of exponential algebraic closure and establishes a weak Schanuel property within exponential fields.

Findings

Exponential algebraic closure operator is a pregeometry.

The dimension function satisfies a weak Schanuel property.

There are at most countably many counterexamples to Schanuel's conjecture.

Abstract

I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property. A corollary is that there are at most countably many essential counterexamples to Schanuel's conjecture.