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

Françoise Point

TL;DR

The paper addresses definable groups inside dp-minimal topological fields expanded by a generic derivation, proving that every finite-dimensional $ abla$-definable group embeds densely and definably into an $ ext{L}$-definable group, with a prolongation-controlled realization $ abla_N( ext{$ abla$-Γ})$ inside. Under dp-minimal or related open/rank-1 hypotheses, it further shows that such groups arise as sharp points of an $ ext{L}$-definable $D$-group, extending Buium’s algebraic $D$-groups to this abstract setting. The work develops an axiomatic framework linking $ abla$-definable groups to purely algebraic definable groups via cell decomposition, prolongations, and Weil pre-group constructions, enabling a transfer of differential-algebraic structure to static definable groups. Together, these results generalize prior PPP-type findings to dp-minimal topological fields and provide a structural bridge between differential expansions and definable group theory with potential impacts on model theory of valued fields and differential-algebraic geometry.

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 $Γ$ be a finite-dimensional $\mathcal{L}_\partial$-definable group in a model of $T_\partial$. Then we show that $Γ$ densely and definably embeds in an $\mathcal{L}$-definable group $G$. Further, using a $C^1$-cell decomposition result, we show that $Γ$ 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.

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

TL;DR

The paper addresses definable groups inside dp-minimal topological fields expanded by a generic derivation, proving that every finite-dimensional -definable group embeds densely and definably into an -definable group, with a prolongation-controlled realization abla inside. Under dp-minimal or related open/rank-1 hypotheses, it further shows that such groups arise as sharp points of an -definable -group, extending Buium’s algebraic -groups to this abstract setting. The work develops an axiomatic framework linking -definable groups to purely algebraic definable groups via cell decomposition, prolongations, and Weil pre-group constructions, enabling a transfer of differential-algebraic structure to static definable groups. Together, these results generalize prior PPP-type findings to dp-minimal topological fields and provide a structural bridge between differential expansions and definable group theory with potential impacts on model theory of valued fields and differential-algebraic geometry.

Abstract

Let be a complete, model-complete, geometric dp-minimal -theory of topological fields of characteristic and let be the theory of expansions of models of by a derivation . We assume that has a model-companion . Let be a finite-dimensional -definable group in a model of . Then we show that densely and definably embeds in an -definable group . Further, using a -cell decomposition result, we show that densely and definably embeds in a definable -group, generalizing the classical construction of Buium of algebraic -groups and extending for that class of fields, results obtained in arXiv:2208.08293, arXiv:2305.16747.
Paper Structure (21 sections, 30 theorems, 61 equations)

This paper contains 21 sections, 30 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric theories
  4. dp-minimal fields
  5. Partial derivatives of correspondences
  6. Implicit function theorems
  7. Differential expansions
  8. Cell decomposition
  9. Open topological fields
  10. Topological groups
  11. $\mathcal{L}_\partial$-Definable sets in models of $T_\partial$
  12. Weil pre-group construction
  13. The construction
  14. $\mathcal{L}_\partial$-definable groups in models of $T_\partial$
  15. Finite-dimensional groups
  16. ...and 6 more sections

Key Result

Proposition 2.9

Let $T$ be dp-minimal and not strongly minimal. Let $V$ be an open definable subset of $K^n$ and let $f: V\rightrightarrows K$ be a definable continuous $m$-correspondence. Then for each $1\leq i\leq n$, $\partial_{x_{i}} f$ exists and is continuous almost everywhere.

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.5
  • Definition 2.7
  • Proposition 2.9
  • Proposition 2.11
  • Claim 2.1
  • Corollary 2.12
  • Definition 2.14
  • Theorem 2.15
  • ...and 64 more