Explicit Factorization of Piecewise Constant Matrix Functions

Within the framework of differential Galois theory, we provide a solution to the factorization problem of piecewise constant matrix functions that arise from Fuchsian systems of differential equations. We calculate the partial indices of these matrix functions using the methods of the Riemann-Hilbert monodromy problem. This allows us to determine the explicit form of the factorization and understand the behavior of these matrix functions in relation to the underlying Fuchsian system of differential equations.