On the generalized Riemann-Hilbert problem with irregular singularities

Andrey Bolibruch (1) , Stéphane Malek (1) , Claude Mitschi (1)

1 IRMA - Institut de Recherche Mathématique Avancée

We give sufficient conditions, on data including the monodromy representation, the Stokes matrices and the Poincaré rank at prescribed singularities, to solve the generalized Riemann-Hilbert problem with irregular singularities. We recover in particular the irreducibility condition on the monodromy given by Bolibrukh and Kostov in the classical case. We apply these criteria to solve the inverse problem in differential Galois theory with a better control of the singularities.

Mots clés: "système différentiel linéaire ordinaire; singularité irrégulière; monodromie; matrice de Stokes; rang de Poincaré; fibré vectoriel; connexion méromorphe; groupe de galois différentiel"; système différentiel linéaire ordinaire; groupe de galois différentiel

Domaines: Equations aux dérivées partielles [math.AP]

Andrey Bolibruch, Stéphane Malek, Claude Mitschi. On the generalized Riemann-Hilbert problem with irregular singularities. 2004. ⟨hal-00129699⟩