An estimate of the number of apparent singularities in the Riemann-Hilbert problem on a compact Riemann surface

TL;DR

This paper provides an upper estimate on the number of apparent singularities needed to construct linear differential equations with specified monodromy and fuchsian singularities on a compact Riemann surface.

Contribution

It introduces a new upper bound estimate for apparent singularities in the Riemann-Hilbert problem on Riemann surfaces.

Findings

Derived an explicit upper estimate for apparent singularities.

Applicable to systems with given monodromy and fuchsian singularities.

Enhances understanding of the Riemann-Hilbert correspondence.

Abstract

In the paper we give an upper estimate of the number of apparent singularities that are sufficient for construction of a system of linear differential equations on a Riemann surface with given fuchsian singularities and monodromy.