Download this article
 Download this article For screen
For printing
Recent Issues

Volume 18
Issue 12, 2133–2308
Issue 11, 1945–2131
Issue 10, 1767–1943
Issue 9, 1589–1766
Issue 8, 1403–1587
Issue 7, 1221–1401
Issue 6, 1039–1219
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN 1944-7833 (online)
ISSN 1937-0652 (print)
 
Author index
To appear
 
Other MSP journals
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

Open Access made possible by participating institutions via Subscribe to Open.