On generating relative and absolute invariants of linear differential equations

TL;DR

This paper presents a method to generate relative and absolute invariants for linear ordinary differential equations using an infinitesimal approach, providing explicit constructions for equations up to fifth order.

Contribution

It introduces a general expression linking relative and absolute invariants and demonstrates how to explicitly generate invariants for all orders, including fifth order equations.

Findings

Derived a general formula for relative invariants.

Explicitly constructed invariants for fifth order linear ODEs.

Showed the effectiveness of the infinitesimal method for invariant generation.

Abstract

A general expression for a relative invariant of a linear ordinary differential equations is given in terms of the fundamental semi-invariant and an absolute invariant. This result is used to established a number of properties of relative invariants, and it is explicitly shown how to generate fundamental sets of relative and absolute invariants of all orders for the general linear equation. Explicit constructions are made for the linear ODE of order five. The approach used for the explicit determination of invariants is based on an infinitesimal method.