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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.

MSC:
 34M99 Ordinary differential equations in the complex domain 12H05 Differential algebra
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References:
 [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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