Groups definable in partial differential fields with an automorphism

TL;DR

This paper investigates the structure of definable groups in partial differential fields with automorphisms, extending known results and identifying key subgroup properties in this mathematical setting.

Contribution

It extends Cassidy's results to partial differential fields with automorphisms, revealing new structural properties of definable groups in this context.

Findings

Zariski dense definable subgroups of simple algebraic groups are characterized.

Existence of a smallest definable subgroup of finite index in these groups.

Analogues of Cassidy's results are established for partial differential fields.

Abstract

In this paper we study groups definable in existentially closed partial differential fields of characteristic 0 with an automorphism which commutes with the derivations. In particular, we study Zariski dense definable subgroups of simple algebraic groups, and show an analogue of Phyllis Cassidy's result for partial differential fields. We also show that these groups have a smallest definable subgroup of finite index.