×

On the definition of the Galois group of linear differential equations. (English. French summary) Zbl 1419.34236

Summary: Let us consider a linear differential equation \(Y' = A Y\) over a differential field \(K\), where \(A \in \mathrm{M}_{\mathrm{n}}(K)\). Let \(F\) be a fundamental system of solutions of the equation. So the differential field extension \(K(F) / K\) depends on the choice of \(F\). We show that Galois group according to the general Galois theory of Umemura is independent of the choice of \(F\) and, in particular, coincides with the Picard-Vessiot Galois group of the equation. Applying this result, we can prove comparison theorems of H. Umemura [Nagoya Math. J. 144, 59–135 (1996; Zbl 0878.12002); Banach Cent. Publ. 94, 263–293 (2011; Zbl 1252.12007)], B. Malgrange [Chin. Ann. Math., Ser. B 23, No. 2, 219–226 (2002; Zbl 1009.12005)] and G. Casale [Sur le groupoïde de Galois d’un feuilletage, PhD thesis, Université Paul Sabatier (2004)].

MSC:

34A30 Linear ordinary differential equations and systems
12H05 Differential algebra
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Casale (G.).— Sur le groupoïde de Galois d’un feuilletage. PhD thesis, Université Paul Sabatier (2004).
[2] Kolchin (E.).— Differential algebra and algebraic groups. Academic Press (1973). · Zbl 0264.12102
[3] Malgrange (B.).— On nonlinear differential Galois theory. Dedicated to the memory of Jacques-Louis Lions. Chinese Annals of Mathematics Series B, 23(2), p. 219-226 (2002). · Zbl 1009.12005
[4] Umemura (H.).— Differential Galois theory of algebraic differential equations. Nagoya Mathematical Journal, 144, p. 1-58 (1996). · Zbl 0885.12004
[5] Umemura (H.).— Differential Galois theory of infinite dimension. Nagoya Mathematical Journal, 144, p. 59-135 (1996). · Zbl 0878.12002
[6] Umemura (H.).— Picard-Vessiot theory in general Galois theory. Algebraic methods in dynamical systems, 94, p. 263-193 (2011). · Zbl 1252.12007
[7] van der Put (M.) and Singer (M. F).— Galois theory of linear differential equations. Springer-Verlag (2003). · Zbl 1036.12008
[8] Zariski (O.) and Samuel (P.).— Commutative Algebra vol. I. Springer-Verlag (1975). · Zbl 0313.13001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.