Strongly Minimal Relics of T-convex Fields

TL;DR

This paper investigates strongly minimal relics in T-convex fields, establishing their dimensionality, exploring the trichotomy reduction, and introducing differentiable Hausdorff geometric fields for a unified framework.

Contribution

It extends the analysis of strongly minimal relics to T-convex fields, proves trichotomy implications, and introduces differentiable Hausdorff geometric fields for broader applicability.

Findings

Non-locally modular strongly minimal relics are two-dimensional.

Trichotomy for definable relics implies trichotomy for interpretable relics.

One-dimensional relics in differentiable Hausdorff geometric fields follow a specific trichotomy.

Abstract

Generalizing previous work on algebraically closed valued fields (ACVF) and o-minimal fields, we study strongly minimal relics of real closed valued fields (RCVF), and more generally T-convex expansions of o-minimal fields. Our main result (replicating the o-minimal setting) is that non-locally modular strongly minimal definable relics of T-convex fields must be two-dimensional. We also continue our work on reducing the trichotomy for general relics of a structure to just the relics of certain distinguished sorts. To this end, we prove that the trichotomy for definable RCVF-relics implies the trichotomy for interpretable RCVF-relics, and also that the trichotomy for relics of o-minimal fields implies the trichotomy for relics of any dense o-minimal structure. Finally, we introduce the class of differentiable Hausdorff geometric fields (containing o-minimal fields and various valued fields), and give a general treatment of the trichotomy for one-dimensional relics of such fields (namely, reducing the trichotomy for one-dimensional relics to an axiomatic condition on the field itself).