On sufficient conditions for the existence of a Fuchsian equation with prescribed monodromy
The paper offers an excellent review of some known results regarding sufficient conditions for positive solvability of the Riemann-Hilbert problem. It then goes into some new results regarding the solvability of the problem. The paper addresses the Riemann-Hilbert problem for a certain class of linear ordinary differential equations in the complex domain. The system \[ {dy\over dz}= B(z)y \] with unknown vector-valued function, \(y= (y',\dots, y^p)^t\), (where \(t\) is transposition), has singularities at \(a_1,\dots, a_n\). That is to say \(B(z)dz\) is holomorphic in \(\mathbb{C}\setminus\{a_1,\dots, a_n\}\) and \(\mathbb{C}\) is the Riemann sphere. Should the \(a_i\) be a pole of order one the system is said to be Fuchsian. Then the 21st Hilbert problem for Fuchsian linear systems is formulated and very well treated in the paper. Several excellent results are obtained regarding the form of all the monodromy matrices for the problem.
scientific article; zbMATH DE number 1431889