Factorization of piecewise constant matrix functions and systems of linear differential equations

TL;DR

This paper investigates the factorization of piecewise constant matrix functions on complex curves, reducing the problem to linear differential systems, and provides explicit solutions for specific cases with applications to monodromy of Fuchsian equations.

Contribution

It introduces a reduction of the matrix function factorization problem to linear differential equations and solves it explicitly for certain cases, advancing understanding of partial indices and monodromy.

Findings

Complete solutions for n=2, m=4 and n=m=3 cases.

Explicit determination of partial indices in some cases.

Reduction of the factorization problem to monodromy of Fuchsian equations.

Abstract

Let G be a piecewise constant $n\times n$ matrix function which is defined on a smooth closed curve $\Gamma$ in the complex sphere and which has m jumps. We consider the problem of determining the partial indices of the factorization of the matrix function G in the space $L^p(\Gamma)$. We show that this problem can be reduced to a certain problem for systems of linear differential equations. Studying this related problem, we obtain some results for the partial indices for general n and m. A complete answer is given for n=2, m=4 and for n=m=3. One has to distinguish several cases. In some of these cases, the partial indices can be determined explicitly. In the remaining cases, one is led to two possibilities for the partial indices. The problem of deciding which is the correct possibility is equivalent to the description of the monodromy of n-th order linear Fuchsian differential equations with m singular points.