$T$-convex $T$-differential fields and their immediate extensions

TL;DR

This paper studies $T$-convex $T$-differential fields within polynomially bounded o-minimal theories, proving the existence of immediate spherically complete extensions and exploring relaxations of boundedness assumptions.

Contribution

It establishes the existence of immediate spherically complete extensions for $T$-convex $T$-differential fields and extends results to cases with relaxed boundedness conditions.

Findings

Every $T$-convex $T$-differential field has an immediate spherically complete extension.

The assumption of polynomial boundedness can be relaxed to power boundedness in key cases.

Extensions preserve $T$-convexity and $T$-differential structure.

Abstract

Let $T$ be a polynomially bounded o-minimal theory extending the theory of real closed ordered fields. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring and a $T$-derivation. If this derivation is continuous with respect to the valuation topology, then we call $K$ a $T$-convex $T$-differential field. We show that every $T$-convex $T$-differential field has an immediate strict $T$-convex $T$-differential field extension which is spherically complete. In some important cases, the assumption of polynomial boundedness can be relaxed to power boundedness.