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Galois groups, Stokes operators and a theorem of Ramis. (English. Russian original) Zbl 0719.34013
Funct. Anal. Appl. 24, No. 4, 286-296 (1990); translation from Funkts. Anal. Prilozh. 24, No. 4, 31-42 (1990).
Let F be a field of germs of meromorphic functions at the origin, \(A\in M_ F(n)\) and (1) \(y'=Ay\) be a system of differential equations. J.-P. Ramis [C. R. Acad. Sci., Paris, Sér. I 301, 99-102 (1985; Zbl 0582.34006); ibid. 165-167 (1985; Zbl 0593.12005)] gave a sketch of proof of the following
Theorem. The Galois group of system (1) over F contains the Stokes transformations.
In the paper the authors obtain an alternative proof of the theorem which is based on a calculation of functional cochains. They advance also the hypothesis that the subgroup generated by Stokes and monodromy transformations is dense in the Galois group.

34M99 Ordinary differential equations in the complex domain
12H05 Differential algebra
Full Text: DOI
[1] J. P. Ramis, ”Phénomène de Stokes et resommation,” C. R. Acad. Sci. Paris Sér. I Math.,301, No. 4, 99-102 (1985). · Zbl 0582.34006
[2] J. P. Ramis, ”Phénomène de Stokes et filtration de Gevrey sur le group de Picard-Vessiot,” C. R. Acad. Sci. Paris Sér. I Math., No. 5, 165-167 (1985). · Zbl 0593.12015
[3] Yu. S. Il’yashenko, ”Finiteness theorems for limit cycles I,” Usp. Mat. Nauk,45, No. 2, 143-200 (1990).
[4] J. Écalle, J. Martinet, R. Moussu, and J. P. Ramis, ”Non-accumulation de cycles limites,” C. R. Acad. Sci. Paris Sér. I Math.,307, No. 14, 375-378, 431-434 (1987). · Zbl 0615.58011
[5] V. I. Arnol’d and Yu. S. Il’yashenko, ”Ordinary differential equations,” in Contemporary Problems of Mathematics. Fundamental Directions. Dynamic Systems [in Russian], VINITI, Moscow (1985).
[6] M. Jurkat, Meromorphe Differentialgleihungen, Lecture Notes in Math., No. 637, Springer-Verlag, Berlin (1978).
[7] R. Courant, Geometric Theory of Functions of a Complex Variable [Russian translation], ONTI, Moscow?Leningrad (1934).
[8] E. C. Titchmarsh, The Theory of Functions, Clarendon Press, Oxford (1932). · Zbl 0005.21004
[9] A. G. Khovanskii, ”The representability of functions in quadratures,” Usp. Mat. Nauk,26, No. 4, 251-252 (1971). · Zbl 0235.30007
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