The uniform companion for fields with free operators in characteristic zero
Shezad Mohamed
TL;DR
The paper extends the uniform companion paradigm from differential fields to fields with free operators, constructing UC_{\mathcal{D}} for characteristic 0 and showing the model companion behavior when the associated difference theory is difference large and model complete. It proves that, under a local assumption on {\mathcal{D}}, simplicity transfers from the base field theory to the {\mathcal{D}}-field theory, enabling a detailed analysis of bounded, PAC {\mathcal{D}}-fields and their imaginaries. It provides alternative geometric characterizations via D-varieties, establishes transfer results for NIP and stability, and develops pseudo {\mathcal{D}}-closed fields with a robust notion of independence. The non-local case clarifies the necessity of locality: for base fields finitely generated over {\mathbb{Q}}, the uniform companion exists exactly when {\mathcal{D}} is local, with model companions failing in many non-local scenarios such as Th({\mathbb{Q}}_{p}) with {\mathcal{D}}-fields. The work thus unifies differential and difference perspectives, offering a broad framework for fields equipped with free operator structures.
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.