On the Riemann–Hilbert Problem in Dimension 4
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We consider the problem of existence of a Fuchsian system with a prescribed 4-dimensional monodromy. We give a classification of all cases of negative solution of this problem in terms of reducibility pattern of the representation, its local structure (which is described by a modification of Jordan form), and restrictions on asymptotics of solutions to Fuchsian systems in lower dimensions. We also show that realization of a reducible 4-dimensional representation by a Fucshian system, if it exists, can be chosen in a block upper-triangular form (though not necessarily with the same reducibility pattern). At the end of the paper, we present new counterexamples to the Riemann–Hilbert problem in dimension 4.
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- 1.A. A. Bolibruch, The Riemann–Hilbert problem. [Russian] Russ. Math. Surv. 45(1990), No. 2(272).Google Scholar
- 2.D.V. Anosov and A.A. Bolibruch, The Riemann-Hilbert problem. Aspects Math. Vol. E 22, Braunschweig, Wiesbaden: Vieweg, 1994.Google Scholar
- 3.A. H. M. Levelt, Hypergeometric functions. Nederl. Akad. Wetensch. Proc., Ser. A 64(1961).Google Scholar
- 4.V.P. Kostov, Quantum states of monodromy groups. J. Dynam. Control Syst. 5(1999), No. 1, 51-100.Google Scholar
- 5.______, Fuchsian linear systems on ℂℙ1and the Riemann–Hilbert problem. C.R. Acad. Sci. Paris, Ser. 1 (1992), 143-148.Google Scholar
- 6.Vandamme Juliet, Simplification du thèoreme de Moser. In: Problème de Riemann–Hilbert pour une monodromie rèsoluble de dimension4, Universitède NICE, Prèpublication, No. 482, Mars 1997.Google Scholar
- 7.A. A. Bolibruch, On the problem of existence of Fuchsian systems with given asymptotics. Proc. Steklov Inst. Math. 216(1996), 32-44.Google Scholar
- 8.A. I. Gladyshev, On reducible Fuchsian systems of order 4. [Russian] In: Problems of Mathematics in Physical and Engeneering problems–a collection of articles, Moscow Institute of Physics and Technology, 1994.Google Scholar