Some addition to the generalized Riemann-Hilbert problem

R.R. Gontsov[1]; I.V. Vyugin[1]

  • [1] Institute for Information Transmission Problems, Moscow, Russia

Annales de la faculté des sciences de Toulouse Mathématiques (2009)

  • Volume: 18, Issue: 3, page 561-576
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Gontsov, R.R., and Vyugin, I.V.. "Some addition to the generalized Riemann-Hilbert problem." Annales de la faculté des sciences de Toulouse Mathématiques 18.3 (2009): 561-576. <http://eudml.org/doc/10117>.

@article{Gontsov2009,
abstract = {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.},
affiliation = {Institute for Information Transmission Problems, Moscow, Russia; Institute for Information Transmission Problems, Moscow, Russia},
author = {Gontsov, R.R., Vyugin, I.V.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {monodromy; irregular singularity; Poincaré rank; generalized Riemann-Hilbert problem; holomorphic vector bundle},
language = {eng},
month = {7},
number = {3},
pages = {561-576},
publisher = {Université Paul Sabatier, Toulouse},
title = {Some addition to the generalized Riemann-Hilbert problem},
url = {http://eudml.org/doc/10117},
volume = {18},
year = {2009},
}

TY - JOUR
AU - Gontsov, R.R.
AU - Vyugin, I.V.
TI - Some addition to the generalized Riemann-Hilbert problem
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2009/7//
PB - Université Paul Sabatier, Toulouse
VL - 18
IS - 3
SP - 561
EP - 576
AB - 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.
LA - eng
KW - monodromy; irregular singularity; Poincaré rank; generalized Riemann-Hilbert problem; holomorphic vector bundle
UR - http://eudml.org/doc/10117
ER -

References

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  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). Zbl0446.34007MR597121
  5. Bolibruch (A. A.).— Vector bundles associated with monodromies and asymptotics of Fuchsian systems. J. Dynam. Control Systems1:2, p. 229-252 (1995). Zbl0943.34079MR1333772
  6. Bolibruch (A. A.), Malek (S.), Mitschi (C.).— On the generalized Riemann-Hilbert problem with irregular singularities. Exposition. Math.24, p. 235-272 (2006). Zbl1106.34061MR2250948
  7. Deligne (P.).— Equations différentielles à points singuliers réguliers. Lecture Notes in Math.163, Springer-Verlag, Berlin, (1970). Zbl0244.14004MR417174
  8. Forster (O.).— Riemannsche Flachen. Springer-Verlag, Berlin, (1980). Zbl0381.30021MR447557
  9. Hartman (P.).— Ordinary differential equations. Wiley, New York, (1964). Zbl0125.32102MR171038
  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). Zbl0577.34029MR815250
  11. V’yugin (I. V.), Gontsov (R. R.).— Additional parameters in inverse monodromy problems. Sbornik: Mathematics197:12, p. 1753-1773 (2006). Zbl1159.34059MR2437080

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