Differential-henselianity and maximality of asymptotic valued differential fields

TL;DR

This paper proves that asymptotic valued differential fields have unique maximal immediate extensions and that differential-henselian asymptotic fields are differential-algebraically maximal, establishing existence and uniqueness of their differential-henselizations.

Contribution

It removes the monotonicity assumption in the characterization of differential-henselian asymptotic fields and proves the existence and uniqueness of their differential-henselizations.

Findings

Unique maximal immediate extensions of asymptotic valued differential fields.

Differential-henselian asymptotic fields are differential-algebraically maximal.

Existence and uniqueness of differential-henselizations for asymptotic fields.

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 (arXiv:1509.02588, Theorem 7.0.3). Finally, we use this result to show the existence and uniqueness of differential-henselizations of asymptotic fields.