Solvability of Equations by Quadratures and Newton's Theorem

TL;DR

This paper demonstrates that an elementary method based on Puiseux series can determine the solvability of linear differential equations by quadratures for any order, simplifying the complex Galois-theoretic approach.

Contribution

It extends Liouville and Ritt's elementary criteria to arbitrary order using Puiseux series, providing a simpler proof for solvability conditions.

Findings

Elementary Puiseux series method works for all orders

Simplifies the proof of solvability criteria

Matches the conditions given by Picard--Vessiot theorem

Abstract

Picard--Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order $n$ by quadratures in terms of its Galois group. It is based on the differential Galois theory and is rather involved. J.Liouville in 1839 found an elementary criterium for such solvability for $n=2$. J.F.Ritt simplified Liouville's theorem (1948). In 1973 M. Rosenlicht proved a similar criterium for arbitrary $n$. Rosenlicht work relies on the valuation theory and is not elementary. In these notes we show that the elementary Liouville--Ritt method based on developing solutions in Puiseux series as functions of a parameter works smoothly for arbitrary $n$ and proves the same criterium.