On the Deligne-Simpson problem and its weak version

We consider the Deligne-Simpson problem (DSP) (respectively the weak DSP): Give necessary and sufficient conditions upon the choice of the p + 1 conjugacy classes c(j) subset of gl(n, C) or C-j subset of GL(n, C) so that there exist irreducible (p + 1)-tuples (respectively (p + 1)-tuples with trivial centralizers) of matrices A(j) is an element of c(j) with zero sum or of matrices M-j is an element of C-j whose product is I. The matrices A(j) (respectively M-j) are interpreted as matrices-residua of Fuchsian linear systems (respectively as monodromy matrices of regular linear systems) of differential equations with complex time. In the paper we give sufficient conditions for solvability of the DSP in the case when one of the matrices is with distinct eigenvalues. (C) 2004 Elsevier SAS. All rights reserved.