Monotone $T$-convex $T$-differential fields

TL;DR

This paper develops a model theory for monotone $T$-convex $T$-differential fields, proving the existence of a complete, distal model completion and introducing a $T^{oundary}$-henselianity concept, extending valued differential field theory.

Contribution

It establishes a model completion for monotone $T$-convex $T$-differential fields, including a new $T^{oundary}$-henselianity axiom and an Ax--Kochen/Ershov theorem for these structures.

Findings

The theory of monotone $T$-convex $T$-differential fields has a model completion.

The model completion is complete and distal.

A $T^{oundary}$-henselianity condition is introduced and analyzed.

Abstract

Let $T$ be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that $T$ is power bounded. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring $\mathcal{O}$ and a $T$-derivation $\partial$ such that $\partial$ is monotone, i.e., weakly contractive with respect to the valuation induced by $\mathcal{O}$. We show that the theory of monotone $T$-convex $T$-differential fields, i.e., the common theory of such $K$, has a model completion, which is complete and distal. Among the axioms of this model completion, we isolate an analogue of henselianity that we call $T^{\partial}$-henselianity. We establish an Ax--Kochen/Ershov theorem and further results for monotone $T$-convex $T$-differential fields that are $T^{\partial}$-henselian.