On the uniqueness of maximal immediate extensions of valued differential fields

TL;DR

This paper investigates the conditions under which maximal immediate extensions of valued differential fields are unique, removing previous assumptions and establishing new existence and uniqueness results for differential-henselizations.

Contribution

It generalizes prior results by replacing monotonicity with a condition on the value group's structure, proving existence and uniqueness of differential-henselizations under these new assumptions.

Findings

Removed the monotonicity assumption in uniqueness results.

Proved existence and uniqueness of differential-henselizations for certain valued differential fields.

Extended the understanding of the relationship between differential-algebraic maximality and differential-henselianity.

Abstract

So far there exist just a few results about the uniqueness of maximal immediate valued differential field extensions and about the relationship between differential-algebraic maximality and differential-henselianity; see arXiv:1509.02588, Chapter 7. We remove here the assumption of monotonicity in these results but replace it with the assumption that the value group is the union of its convex subgroups of finite (archimedean) rank. We also show the existence and uniqueness of differential-henselizations of asymptotic fields with such a value group.