Model theory of fields with free operators in characteristic zero

TL;DR

This paper generalizes the model theory of fields with operators, establishing the existence of a model companion for fields with free operators over characteristic zero fields, unifying difference and differential fields.

Contribution

It introduces a unified framework for fields with free operators, proving the existence of a simple model companion satisfying the Zilber dichotomy in characteristic zero.

Findings

Existence of a model companion for ${ m D}$-fields in characteristic zero.

The ${ m D}$-CF$_0$ theory is simple.

The theory satisfies the Zilber dichotomy for finite-dimensional minimal types.

Abstract

Generalising and unifying the known theorems for difference and differential fields, it is shown that for every finite free ${\mathbb S}$-algebra ${\mathcal D}$ over a field $A$ of characteristic zero the theory of ${\mathcal D}$-fields has a model companion ${\mathcal D}$-CF$_0$ which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.