Schwarz maps of Algebraic Linear Ordinary Differential Equations

TL;DR

This paper investigates algebraic linear ordinary differential equations, providing methods to find closed-form solutions and classify such equations with rational coefficients using Schwarz maps and Picard-Vessiot theory.

Contribution

It introduces a novel approach to solving algebraic linear ODEs and classifies them via Schwarz maps, advancing the understanding of their structure and solutions.

Findings

Closed-form solutions for algebraic linear ODEs derived

Method to classify algebraic ODEs with rational coefficients developed

Schwarz map analysis linked to Picard-Vessiot theory

Abstract

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.