Some addition to the generalized Riemann-Hilbert problem

Abstract

We consider the generalized Riemann-Hilbert problem for linear differential equations with irregular singularities. After recalling the formulation of the problem in terms of vector bundles over the Riemann sphere, we give some estimates for the unique non-minimal Poincaré rank of the system and the number of apparent singularities of the scalar equation constructed by corresponding generalized monodromy data.

Andrei Andreevich Bolibrukh's works on the analytic theory of differential equations

“…The second type of local problems for irregular systems, which have a long history of investigations, is the Birkhoff problem, which we discuss in the next section. Global problems connected with the generalized Riemann-Hilbert problem and differential Galois theory have relatively recently come under scrutiny (see [3], [47], [48], and references in these papers). Bolibrukh took the initiative in investigating the generalized Riemann-Hilbert problem in [47], where several sufficient conditions for the positive solvability of the problem were obtained.…”

Различные Варианты Проблемы Римана - Гильберта Для Линейных Дифференциальных Уравнений

“…(Два линейных дифференциальных уравнения будем называть мероморфно эквивалентными в окрестности особой точки, если ме-роморфно эквивалентны соответствующие им линейные системы с матрицами коэффициентов вида (28) (31); см. [57]. Заметим, что в случае, когда все локальные уравнения (31) фуксовы, зада-ча превращается в классическую проблему Римана-Гильберта для скалярных фуксовых уравнений.…”