On definable groups in dp-minimal topological fields equipped with a generic derivation

TL;DR

This paper proves that in certain dp-minimal topological fields with a generic derivation, definable groups can be densely embedded into definable algebraic D-groups, extending classical constructions and previous results.

Contribution

It establishes that definable groups in these fields can be embedded into definable D-groups, generalizing Buium's algebraic D-groups and extending prior work.

Findings

Definable groups embed densely into definable G.

Embedding into D-groups using $C^1$-cell decomposition.

Extension of classical algebraic D-group constructions.

Abstract

Let $T$ be a complete, model-complete, geometric dp-minimal $\mathcal{L}$-theory of topological fields of characteristic $0$ and let $T(\partial)$ be the theory of expansions of models of $T$ by a derivation $\partial$. We assume that $T(\partial)$ has a model-companion $T_{\partial}$. Let $\Gamma$ be a finite-dimensional $\mathcal{L}_\partial$-definable group in a model of $T_\partial$. Then we show that $\Gamma$ densely and definably embeds in an $\mathcal{L}$-definable group $G$. Further, using a $C^1$-cell decomposition result, we show that $\Gamma$ densely and definably embeds in a definable $D$-group, generalizing the classical construction of Buium of algebraic $D$-groups and extending for that class of fields, results obtained in arXiv:2208.08293, arXiv:2305.16747.