Liouvillian Solutions of First Order Non Linear Differential Equations

TL;DR

This paper investigates Liouvillian solutions of first order nonlinear differential equations, proving the existence of specific elements satisfying linear equations within intermediate fields, and offers new insights and proofs related to classical results in the field.

Contribution

It establishes a new property of elements in Liouvillian extensions and generalizes previous results on solutions of nonlinear differential equations.

Findings

Existence of elements satisfying linear equations in intermediate fields

Generalizations of classical results by Singer and Rosenlicht

New proofs for known properties of Liouvillian solutions

Abstract

Let $k$ be a differential field of characteristic zero and $E$ be a liouvillian extension of $k$. For any differential subfield $K$ intermediate to $E$ and $k$, we prove that there is an element in the set $K-k$ satisfying a linear homogeneous differential equation over $k$. We apply our results to study liouvillian solutions of first order non linear differential equations and provide generalisations and new proofs for several results of M. Singer and M. Rosenlicht on this topic.