# Generalized Picard-Vessiot extensions and differential Galois cohomology

## TL;DR

This paper investigates the extension of Picard-Vessiot theory to multiple derivations and automorphisms, providing counterexamples and some positive results in the context of differential Galois cohomology.

## Contribution

It introduces the concept of generalized Picard-Vessiot extensions for multiple derivations and automorphisms, and explores their properties with new counterexamples and positive results.

## Key findings

- Counterexamples for generalized Picard-Vessiot extensions with multiple derivations and automorphisms.
- Positive results for certain cases with multiple derivations.
- Extension of differential Galois cohomology concepts to broader settings.

## Abstract

In an earlier paper it was proved that if a differential field $(K,\delta)$ is algebraically closed and closed under Picard-Vessiot extensions then every differential algebraic principal homogeneous space over K for a linear differential algebraic group G over K has a K-rational point (in fact if and only if). This paper explores whether and if so, how, this can be extended to (a) several derivations, (b) one automorphism. Under a natural notion of "generalized Picard-Vessiot extension" (in the case of several derivations), we give a counterexample. We also have a counterexample in the case of one automorphism. We also formulate and prove some positive statements in the case of several derivations.