Model theory of fields with free operators in characteristic zero

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.