Calculating Galois groups of third order linear differential equations with parameters

TL;DR

This paper introduces the first algorithm for computing the Galois group of third order parameterized linear differential equations, advancing the understanding of their symmetries and solution structures.

Contribution

It provides a novel algorithm specifically designed for third order equations with parameters, filling a gap in differential Galois theory.

Findings

Algorithm successfully computes Galois groups for third order equations.

Extends methods used for second order equations to third order cases.

Facilitates analysis of hypertranscendence in solutions.

Abstract

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel's equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In the present paper, we give the first known algorithm that calculates the differential Galois group of a third order parameterized linear differential equation.