Picard-Vessiot Extensions of Real Differential Fields

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2019
• Main Authors: Crespo, T., Hajto, Z.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2019
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210288
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Picard-Vessiot Extensions of Real Differential Fields / T. Crespo, Z. Hajto // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 27 назв. — англ.

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 that 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(√-1). 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.