Picard-Vessiot Extensions of Real Differential Fields

For a linear differential equation defined over a formally real differential field K with real closed field of constants k, Crespo, Hajto and van der Put proved that there exists a unique formally real Picard- Vessiot extension up to K-differential automorphism. However such an equation may have Picard-Vessiot extensions which are not formally real fields. The differential Galois group of a Picard-Vessiot extension for this equation has the structure of a linear algebraic group defined over k and is a k-form of the differential Galois group H of the equation over the differential field K(i), where i denotes a square root of -1 in the algebraic closure of k. These facts lead us to consider two issues: determining the number of K-differential isomorphism classes of Picard-Vessiot extensions and describing the variation of the differential Galois group in the set of k-forms of H. We address these two issues in the cases when H is a special linear, a special orthogonal, or a symplectic linear algebraic group and conclude that there is no general behaviour

Galois correspondence theorem for Picard-Vessiot extensions

In this paper, we generalize the definition of the differential Galois group and the Galois correspondence theorem established previously for Picard-Vessiot extensions of real differential fields with real closed field of constants to any Picard-Vessiot extension.Comment: arXiv admin note: substantial text overlap with arXiv:1207.189