Differentially large fields

León Sánchez, Omar; Tressl, Marcus

doi:10.2140/ant.2024.18.249

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Differentially large fields

Omar León Sánchez and Marcus Tressl

Vol. 18 (2024), No. 2, 249–280

DOI: 10.2140/ant.2024.18.249

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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Vol. 18, No. 2, 2024