On definable groups in real closed fields with a generic derivation, and related structures

TL;DR

This paper investigates groups definable in real closed fields with a generic derivation, showing they embed into semialgebraic groups and extending results to broader geometric and topological contexts.

Contribution

It proves that finite-dimensional definable groups in CODF embed into semialgebraic groups and generalizes this to various geometric and topological theories.

Findings

Definable groups in CODF embed into semialgebraic groups

Extension of results to large geometric fields with generic derivation

General theorem on recovering groups from generic data

Abstract

We study finite-dimensional groups definable in models of the theory of real closed fields with a generic derivation (also known as CODF). We prove that any such group definably embeds in a semialgebraic group. We extend the results to several more general contexts; strongly model complete theories of large geometric fields with a generic derivation, model complete o-minimal expansions of RCF with a generic derivation, open theories of topological fields with a generic derivation. We also give a general theorem on recovering a definable group from generic data in the context of geometric structures.