# Existentially Closed Exponential Fields

## TL;DR

This paper characterizes the models of exponential fields that are existentially closed, explores their logical properties using positive logic, and situates them within the stability hierarchy, revealing complex independence and amalgamation properties.

## Contribution

It provides a novel characterization of existentially closed exponential fields and extends classification theory concepts to positive logic.

## Key findings

- Existentially closed exponential fields do not form an elementary class.
- They are NSOP$_1$ but TP$_2$ in the stability hierarchy.
- The paper develops a notion of independence and describes amalgamation in these fields.

## Abstract

We characterise the existentially closed models of the theory of exponential fields. They do not form an elementary class, but can be studied using positive logic. We find the amalgamation bases and characterise the types over them. We define a notion of independence and show that independent systems of higher dimension can also be amalgamated. We extend some notions from classification theory to positive logic and position the category of existentially closed exponential fields in the stability hierarchy as NSOP$_1$ but TP$_2$.