Liouville closed $H_T$-fields

TL;DR

This paper investigates $H_T$-fields, models of o-minimal theories with derivations, showing that under power boundedness, such fields have at most two minimal Liouville closed extensions, with some cases allowing exponential functions.

Contribution

It establishes the uniqueness and number of minimal Liouville closed $H_T$-field extensions for power bounded o-minimal theories, extending to certain exponential cases.

Findings

$H_T$-fields have at most two minimal Liouville closed extensions.

Power boundedness condition is crucial for the main result.

Results extend to some exponential o-minimal theories.

Abstract

Let $T$ be an o-minimal theory extending the theory of real closed ordered fields. An $H_T$-field is a model $K$ of $T$ equipped with a $T$-derivation such that the underlying ordered differential field of $K$ is an $H$-field. We study $H_T$-fields and their extensions. Our main result is that if $T$ is power bounded, then every $H_T$-field $K$ has either exactly one or exactly two minimal Liouville closed $H_T$-field extensions up to $K$-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as $T = \operatorname{Th}(\mathbb{R}_{\operatorname{an},\exp})$.