Differentially Large Henselian Fields and Taylor Morphisms

Abstract

In this thesis, we present a variety of results on two topics relating to differentially large fields: on generalised functorial versions of the Taylor morphism, and on differentially henselian fields, which are an analogue of differentially large fields in the context of henselian valued fields. The Taylor morphism is a tool used in differential algebra to construct differential morphisms from algebraic morphisms in a uniform way. We observe that this construction has certain functorial properties, and we generalise the notion of a Taylor morphism as a functor which satisfies the same properties. We demonstrate that generalised Taylor morphisms have corresponding applications to differentially large fields, and we also study the structure of these generalised Taylor morphisms in some detail. We give a comprehensive overview of the model-theoretic properties of differentially henselian fields, including generalisations of classical results from valuation theory, e.g. Ax-Kochen/Ershov type results, quantifier elimination for equicharacteristic 0 fields with angular components, stable embeddedness properties, and more. We also prove various characterisations of differentially henselian fields analogous to those for differentially large fields. Finally, we adapt the machinery of the differential Weil descent to prove results about algebraic extensions of differentially henselian fields.

Date of Award	31 Dec 2023
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Omar Leon Sanchez (Supervisor) & Marcus Tressl (Supervisor)