Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 100, 11 pages      arXiv:1403.3226      https://doi.org/10.3842/SIGMA.2019.100

Picard-Vessiot Extensions of Real Differential Fields

Teresa Crespo a and Zbigniew Hajto b
a)  Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
b)  Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Prof. S. Łojasiewicza 6, 30-348 Kraków, Poland

Received July 04, 2019, in final form December 22, 2019; Published online December 24, 2019

Abstract
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 $K\big(\sqrt{-1}\big)$. 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.

Key words: real Picard-Vessiot theory; linear algebraic groups; group cohomology; real forms of algebraic groups.

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