The uniform companion for fields with free operators in characteristic zero

TL;DR

This paper develops a model companion theory for fields of characteristic zero equipped with free operators, generalizing previous work on derivations and difference fields, and explores properties like simplicity and applications to bounded PAC fields.

Contribution

It introduces the theory UC_D for fields with free operators determined by a homomorphism to a tensor product, extending the uniform companion concept to a broader class of operators.

Findings

Constructed the model companion UC_D for fields with free operators.

Showed simplicity transfers from the underlying field to the field with operators when D is local.

Analyzed the model theory of bounded, PAC fields with free operators.

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.