On local Weyl equivalence of higher order Fucshian equations

TL;DR

This paper investigates the local classification of higher order Fuchsian linear differential equations, highlighting the challenges due to the lack of a natural group action and extending known results from first order systems.

Contribution

It provides new results on the local classification of higher order Fuchsian equations, refining the understanding of their equivalence classes beyond classical notions.

Findings

Established results on local classification of higher order Fuchsian equations.

Identified differences from the gauge equivalence of first order systems.

Extended the framework for understanding equation types without a natural group action.

Abstract

We study the local classification of higher order Fuchsian linear differential equations under various refinements of the classical notion of the "type of differential equation" introduced by Frobenius. The main source of difficulties is the fact that there is no natural group action generating this classification. We establish a number of results on higher order equations which are similar but not completely parallel to the known results on local (holomorphic and meromorphic) gauge equivalence of systems of first order equations.