The uniform companion for fields with free operators in characteristic zero

Abstract: Generalising the uniform companion for large fields with a single derivation, we construct a theory $\text{UC}_{\mathcal{D}}$ of fields of characteristic $0$ with free operators -- operators determined by a homomorphism from the field to its tensor product with $\mathcal{D}$, a finite-dimensional $\mathbb{Q}$-algebra -- which is the model companion of any theory of a field with free operators whose associated difference field is difference large and model complete. Under the assumption that $\mathcal{D}$ is a local ring, we show that simplicity is transferred from the theory of the underlying field to the theory of the field with operators, and we use this to study the model theory of bounded, PAC fields with free operators.