On the definition of the Galois group of linear differential equations

TL;DR

This paper proves that the Galois group defined by Umemura's general theory matches the classical Picard-Vessiot Galois group for linear differential equations, extending previous comparison results.

Contribution

It establishes the equivalence of Galois groups in the general and classical frameworks for linear differential equations.

Findings

Galois group according to Umemura's theory coincides with the Picard-Vessiot Galois group.

Generalizes the comparison theorem of Umemura and Casale.

Provides a unified understanding of Galois groups in differential equations.

Abstract

Let us consider a linear differential equation over a differential field K. For a differential field extension L/K generated by a fundamental system of the equation, we show that Galois group according to the general Galois theory of Umemura coincides with the Picard-Vessiot Galois group. This conclusion generalized the comparision theorem of Umemura and Casale.