Differential-henselianity and maximality of asymptotic valued differential fields

Nigel Pynn-Coates

Vol. 308 (2020), No. 1, 161–205

DOI: 10.2140/pjm.2020.308.161

Abstract

We show that asymptotic (valued differential) fields have unique maximal immediate extensions. Connecting this to differential-henselianity, we prove that any differential-henselian asymptotic field is differential-algebraically maximal, removing the assumption of monotonicity from a theorem of Aschenbrenner, van den Dries, and van der Hoeven. Finally, we use this result to show the existence and uniqueness of differential-henselizations of asymptotic fields.

Keywords

valued differential fields, asymptotic fields, immediate extensions, differential-henselianity, differential Newton diagrams

Mathematical Subject Classification 2010

Primary: 12H05, 12J10

Milestones

Received: 7 August 2018

Revised: 25 March 2020

Accepted: 24 April 2020

Published: 3 December 2020