Iterative differential Galois theory: a model theoretic approach

TL;DR

This paper extends differential Galois theory to positive characteristic iterative differential fields using model theory, introducing strongly normal extensions and Galois extensions for iterative logarithmic equations.

Contribution

It generalizes Kolchin's theory to positive characteristic and non-linear cases using a model theoretic framework, defining strongly normal extensions and proving their Galois properties.

Findings

Defined strongly normal extensions for iterative differential fields

Proved these extensions have Galois theory and G-primitive element theorem

Established Galois extensions for iterative logarithmic equations

Abstract

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a G-primitive element theorem holds. In addition, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations.