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

Elliot Kaplan

Vol. 320 (2022), No. 2, 261–298

DOI: 10.2140/pjm.2022.320.261

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.

Keywords

o-minimality, spherical completeness, valued differential fields

Mathematical Subject Classification

Primary: 03C64

Secondary: 12H05, 12J10

Milestones

Received: 21 April 2021

Revised: 27 July 2022

Accepted: 27 September 2022

Published: 15 February 2023