222:) found a counterexample to Plemelj's statement. This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind; Bolibrukh showed that for a given pole configuration certain monodromy groups can be realised by regular, but not by Fuchsian systems. (In 1990 he published the thorough study of the case of regular systems of size 3 exhibiting all situations when such counterexamples exists. In 1978 Dekkers had shown that for systems of size 2 Plemelj's claim is true.
88:. The problem requires the production of n functions of the variable z, regular throughout the complex z-plane except at the given singular points; at these points the functions may become infinite of only finite order, and when z describes circuits about these points the functions shall undergo the prescribed
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In fact it is more appropriate to speak not about differential equations but about linear systems of differential equations: in order to realise any monodromy by a differential equation one has to admit, in general, the presence of additional apparent singularities, i.e. singularities with trivial
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and others started raising doubts about
Plemelj's work. In fact, Plemelj correctly proves that any monodromy group can be realised by a regular linear system which is Fuchsian at all but one of the singular points. Plemelj's claim that the system can be made Fuchsian at the last point as well is
194:, for flat algebraic connections with regular singularities and more generally regular holonomic D-modules or flat algebraic connections with regular singularities on principal G-bundles, in all dimensions. The history of proofs involving a single complex variable is complicated.
330:)).) Parallel to this the Grothendieck school of algebraic geometry had become interested in questions of 'integrable connections on algebraic varieties', generalising the theory of linear differential equations on
96:, but the rigorous proof has been obtained up to this time only in the particular case where the fundamental equations of the given substitutions have roots all of absolute magnitude unity.
481:
Deligne, Pierre (1970). Équations différentielles à points singuliers réguliers. (French) Lecture Notes in
Mathematics, Vol. 163. Springer-Verlag, Berlin-New York, 1970. 133 pp. MR0417174
112:. The theory of linear differential equations would evidently have a more finished appearance if the problem here sketched could be disposed of by some perfectly general method.
238:) showed that for any size, an irreducible monodromy group can be realised by a Fuchsian system. The codimension of the variety of monodromy groups of regular systems of size
548:
Gérard, Raymond (1969). Le problème de
Riemann-Hilbert sur une variété analytique complexe. (French) Ann. Inst. Fourier (Grenoble) 19 (1969), fasc. 2, 1--32. MR0281946
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Röhrl, Helmut (1957). Das
Riemann-Hilbertsche Problem der Theorie der linearen Differentialgleichungen. (German) Math. Ann. 133, 1--25. MR0086958
491:, Interscience Tracts in Pure and Applied Mathematics, vol. 16, New York-London-Sydney: Interscience Publishers John Wiley & Sons Inc.,
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proved a precise
Riemann–Hilbert correspondence in this general context (a major point being to say what 'Fuchsian' means). With work by
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local monodromy. In more modern language, the (systems of) differential equations in question are those defined in the
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published a solution in 1908. This work was for a long time accepted as a definitive solution; there was work of
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Bolibrukh, A.A. (1992), "Sufficient conditions for the positive solvability of the
Riemann-Hilbert problem",
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with one independent variable z, I wish to indicate an important problem one which very likely
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himself may have had in mind. This problem is as follows: To show that there always exists a
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Proof of the existence of linear differential equations having a prescribed monodromic group
38:, concerns the existence of a certain class of linear differential equations with specified
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Akademiya Nauk SSSR I Moskovskoe
Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk
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wrote a monograph summing up his work. A few years later the Soviet mathematician
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at those. A more strict version of the problem requires these singularities to be
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Handbuch der
Theorie der linearen Differentialgleichungen vol. 2, part 2, No. 366
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92:. The existence of such differential equations has been shown to be probable by
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The original problem was stated as follows (English translation from 1902):
632:(1976), "An Overview of Deligne's work on Hilbert's Twenty-First Problem",
137:, i.e. poles of first order (logarithmic poles), including at infinity. A
400:, Aspects of Mathematics, E22, Braunschweig: Friedr. Vieweg & Sohn,
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For
Riemann–Hilbert factorization problems on the complex plane, see
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617:: 307–312, Progr. Math., 37, Birkhäuser Boston, Boston, MA, 1983,
375:: 307–312, Progr. Math., 37, Birkhäuser Boston, Boston, MA, 1983,
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Kostov, Vladimir Petrov (1992), "Fuchsian linear systems on
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Treibich Kohn, Armando. (1983), "Un résultat de
Plemelj.",
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poles which cannot be realised by Fuchsian systems equals
187:. It led to several bijective correspondences known as '
157:, up to equivalence. The fundamental group is actually a
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Bolibrukh, A. A. (1990), "The Riemann-Hilbert problem",
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On linear differential equations with certain properties
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wrong, unless the monodromy is diagonalizable there.
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Treibich Kohn, Armando., "Un résultat de Plemelj.",
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165:. The question is whether the mapping from these
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485:Plemelj, Josip (1964), Radok., J. R. M. (ed.),
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515:Matematicheskie Zametki
315:{\displaystyle 2(n-1)p}
224:Andrey A. Bolibrukh
216:Andrey A. Bolibrukh
184:Riemann–Hilbert problem
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159:free group
30:of the 23
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50:Statement
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106:Poincaré
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