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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
© 2024 The Author(s), under
exclusive license to MSP (Mathematical Sciences Publishers).
Distributed under the Creative Commons
Attribution License 4.0 (CC BY) .
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