Monodromy groups of parameterized linear differential equations with regular singularities

TL;DR

This paper investigates the properties of parameterized linear differential equations with regular singularities, extending classical theorems and solving key problems in the parameterized differential Galois theory.

Contribution

It introduces a notion of regular singularities for parameterized systems, proves an analogue of Schlesinger's theorem, and addresses the parameterized Riemann-Hilbert and inverse problems.

Findings

Established a parameterized version of Schlesinger's theorem.

Solved a special case of the inverse problem in parameterized Picard-Vessiot theory.

Extended the concept of regular singularities to parameterized differential systems.

Abstract

We study the notion of regular singularities for parameterized complex ordinary linear differential systems, prove an analogue of the Schlesinger theorem for systems with regular singularities and solve both a parameterized version of the weak Riemann-Hilbert Problem and a special case of the inverse problem in parameterized Picard-Vessiot theory.