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)].


34A30 Linear ordinary differential equations and systems
12H05 Differential algebra
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