Real Picard-Vessiot theory

TL;DR

This paper extends Picard-Vessiot theory to real differential fields, proving the existence of real Picard-Vessiot extensions and establishing a Galois correspondence in this context.

Contribution

It introduces the concept of real Picard-Vessiot extensions and develops the associated Galois theory for real differential fields with real closed constants.

Findings

Existence of real Picard-Vessiot extensions for linear differential equations over real fields.

Definition of differential Galois group in the real setting.

Galois correspondence theorem for real Picard-Vessiot extensions.

Abstract

The existence of a Picard-Vessiot extension for a homogeneous linear differential equation has been established when the differential field over which the equation is defined has an algebraically closed field of constants. In this paper, we prove the existence of a real Picard-Vessiot extension for a homogeneous linear differential equation defined over a real differential field K with real closed field of constants. We give an adequate definition of the differential Galois group of a Picard- Vessiot extension of a real differential field with real closed field of constants and we prove a Galois correspondence theorem for such a Picard-Vessiot extension.