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Picard-Vessiot theory for real fields. (English) Zbl 1298.12004
Let \(F\) be a differential field of characteristic \(0\) with field of constants \(C\) and (1) \(L(y)=0\) be a linear differential equation over \(F\). In the case where the field \(C\) is not algebraically closed a question of the existence of Picard-Vessiot extension of \(F\) for the equation (1) is poorly understood see M. P. Epstein [Ann. Math. (2) 62, 528–547 (1955; Zbl 0065.27001)], but it is of particular interest, for example, in the study of equations with parameters [P. J. Cassidy and M. F. Singer, in: Differential equations and quantum groups. Andrey A. Bolibrukh memorial volume. IRMA Lect. Math. Theor. Phys. 9, 113–155 (2007; Zbl 1230.12003)] or solution of the equation by real Liouville functions see O. A. Gel’fond and A. G. Khovanskii [Funkts. Anal. Prilozh. 14, No. 2, 52–53 (1980; Zbl 0468.30006)].
In this paper the authors show on concrete examples why it is difficult to investigate the case where \(C\) is not algebraically closed. Then they prove the existence theorem and main theorem of Galois theory when \(F\) is real and \(C\) is real closed.

12H05 Differential algebra
34A30 Linear ordinary differential equations and systems, general
37C10 Dynamics induced by flows and semiflows
Full Text: DOI arXiv
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