Model theory of differential-henselian pre-$H$-fields

TL;DR

This paper develops a model theory for differential-Hensel-Liouville closed pre-$H$-fields, establishing quantifier elimination, model completeness, and stability properties, and demonstrating the theory's dependence on the residue field's theory.

Contribution

It introduces a comprehensive model-theoretic analysis of differential-Hensel-Liouville closed pre-$H$-fields, including quantifier elimination and stability results, which were previously unknown.

Findings

The theory is determined by the residue field's theory.

Quantifier elimination is achieved in a two-sorted setting.

The theory is complete, distal, and locally o-minimal.

Abstract

Pre-$H$-fields are ordered valued differential fields satisfying some basic axioms coming from transseries and Hardy fields. We study pre-$H$-fields that are differential-Hensel-Liouville closed, that is, differential-henselian, real closed, and closed under exponential integration, establishing an Ax--Kochen/Ershov theorem for such structures: the theory of a differential-Hensel-Liouville closed pre-$H$-field is determined by the theory of its ordered differential residue field; this result fails if the assumption of closure under exponential integration is dropped. In a two-sorted setting with one sort for a differential-Hensel-Liouville closed pre-$H$-field and one sort for its ordered differential residue field, we eliminate quantifiers from the pre-$H$-field sort, from which we deduce that the ordered differential residue field is stably embedded and if it has NIP, then so does the two-sorted structure. Similarly, the one-sorted theory of differential-Hensel-Liouville closed pre-$H$-fields with closed ordered differential residue field has quantifier elimination, is the model completion of the theory of pre-$H$-fields with gap~$0$, and is complete, distal, and locally o-minimal.