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Differentially large fields

Omar León Sánchez and Marcus Tressl

Vol. 18 (2024), No. 2, 249–280
Abstract

We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of “tame” differential fields. We state several characterizations and exhibit plenty of examples and applications. Our results strongly indicate that differentially large fields will play a key role in differential field arithmetic. For instance, we characterize differential largeness in terms of being existentially closed in their power series field (furnished with natural derivations), we give explicit constructions of differentially large fields in terms of iterated powers series, we prove that the class of differentially large fields is elementary, and we show that differential largeness is preserved under algebraic extensions, therefore showing that their algebraic closure is differentially closed.

Keywords
differential fields, large fields, Taylor morphism, Picard–Vessiot theory, elimination theory, existentially closed structures
Mathematical Subject Classification
Primary: 12E99, 12H05
Secondary: 03C60, 34M25
Milestones
Received: 21 April 2021
Revised: 14 June 2022
Accepted: 20 March 2023
Published: 6 February 2024
Authors
Omar León Sánchez
Department of Mathematics
University of Manchester
Manchester
United Kingdom
Marcus Tressl
Department of Mathematics
University of Manchester
Manchester
United Kingdom

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