Exponentially Closed Fields and the Conjecture on Intersections with Tori

TL;DR

This paper introduces a new axiomatization of exponentially closed fields, explores their stability properties, and links their elementary class to a major conjecture in diophantine geometry regarding intersections with tori.

Contribution

It provides an axiomatization of exponentially closed fields and establishes a connection between their elementary class and the CIT conjecture on tori intersections.

Findings

ECF is superstable over its arithmetic interpretation.

ECF matches the elementary class of pseudo-exponential fields if CIT is true.

The paper links model theory of exponential fields to diophantine conjectures.

Abstract

We give an axiomatization of the class ECF of exponentially closed fields, which includes the pseudo-exponential fields previously introduced by the second author, and show that it is superstable over its interpretation of arithmetic. Furthermore, ECF is exactly the elementary class of the pseudo-exponential fields if and only if the diophantine conjecture CIT on atypical intersections of tori with subvarieties is true.