We consider the generalized Riemann-Hilbert problem for linear differential equations with irregular singularities. If one weakens the conditions by allowing one of the Poincaré ranks to be non-minimal, the problem is known to have a solution. In this article we give a bound for the possibly non-minimal Poincaré rank. We also give a bound for the number of apparent singularities of a scalar equation with prescribed generalized monodromy data.
Nous considérons le problème de Riemann-Hilbert généralisé pour des équations différentielles linéaires avec singularités irrégulières. Si on affaiblit les conditions en autorisant que l’un des rangs de Poincaré ne soit pas minimal, il est connu que le problème a une solution. Dans cet article nous donnons une borne pour le rang de Poincaré ainsi obtenu. Nous donnons aussi une borne pour le nombre de singularités apparentes de l’équation scalaire avec une donnée de monodromie généralisée prescrite.
@article{AFST_2009_6_18_3_561_0,
author = {R.R. Gontsov and I.V. Vyugin},
title = {Some addition to the generalized {Riemann-Hilbert} problem},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {561--576},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 18},
number = {3},
year = {2009},
doi = {10.5802/afst.1214},
zbl = {1200.34110},
mrnumber = {2582441},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1214/}
}
TY - JOUR TI - Some addition to the generalized Riemann-Hilbert problem JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2009 DA - 2009/// SP - 561 EP - 576 VL - Ser. 6, 18 IS - 3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1214/ UR - https://zbmath.org/?q=an%3A1200.34110 UR - https://www.ams.org/mathscinet-getitem?mr=2582441 UR - https://doi.org/10.5802/afst.1214 DO - 10.5802/afst.1214 LA - en ID - AFST_2009_6_18_3_561_0 ER -
%0 Journal Article %T Some addition to the generalized Riemann-Hilbert problem %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2009 %P 561-576 %V Ser. 6, 18 %N 3 %I Université Paul Sabatier, Toulouse %U https://doi.org/10.5802/afst.1214 %R 10.5802/afst.1214 %G en %F AFST_2009_6_18_3_561_0
R.R. Gontsov; I.V. Vyugin. Some addition to the generalized Riemann-Hilbert problem. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 3, pp. 561-576. doi : 10.5802/afst.1214. https://afst.centre-mersenne.org/articles/10.5802/afst.1214/
[1] Anosov (D. V.) and Bolibruch (A. A.).— The Riemann-Hilbert problem. Aspects Math. E 22, Friedr. Vieweg & Sohn, Braunschweig, (1994). | MR: 1276272 | Zbl: 0801.34002
[2] Balser (W.).— Birkhoff’s reduction problem. Russian Math. Surveys 59:6, p. 1047-1059 (2004). | MR: 2138466 | Zbl: 1077.34001
[3] Balser (W.), Jurkat (W. B.), Lutz (D. A.).— A general theory of invariants for meromorphic differential equations. I. Formal invariants. Funk. Ekvac. 22:2, p. 197-221 (1979). | MR: 556577 | Zbl: 0434.34002
[4] Balser (W.), Jurkat (W. B.), Lutz (D. A.).— Invariants for redicible systems of meromorphic differential equations. Proc. Edinburgh Math. Soc. 23:2, p. 163-186 (1980). | MR: 597121 | Zbl: 0446.34007
[5] Bolibruch (A. A.).— Vector bundles associated with monodromies and asymptotics of Fuchsian systems. J. Dynam. Control Systems 1:2, p. 229-252 (1995). | MR: 1333772 | Zbl: 0943.34079
[6] Bolibruch (A. A.), Malek (S.), Mitschi (C.).— On the generalized Riemann-Hilbert problem with irregular singularities. Exposition. Math. 24, p. 235-272 (2006). | MR: 2250948 | Zbl: 1106.34061
[7] Deligne (P.).— Equations différentielles à points singuliers réguliers. Lecture Notes in Math. 163, Springer-Verlag, Berlin, (1970). | MR: 417174 | Zbl: 0244.14004
[8] Forster (O.).— Riemannsche Flachen. Springer-Verlag, Berlin, (1980). | MR: 447557 | Zbl: 0381.30021
[9] Hartman (P.).— Ordinary differential equations. Wiley, New York, (1964). | MR: 171038 | Zbl: 0125.32102
[10] Lutz (D. A.), Schäfke (R.).— On the identification and stability of formal invariants for singular differential equations. Linear Algebra Appl. 72, p. 1-46 (1985). | MR: 815250 | Zbl: 0577.34029
[11] V’yugin (I. V.), Gontsov (R. R.).— Additional parameters in inverse monodromy problems. Sbornik: Mathematics 197:12, p. 1753-1773 (2006). | MR: 2437080 | Zbl: 1159.34059
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