Generalized rational first integrals of analytic differential systems

TL;DR

This paper investigates the necessary conditions for the existence of generalized rational first integrals in analytic differential systems, extending classical results and emphasizing the role of resonances and lowest order terms.

Contribution

It extends previous classical and modern results on rational first integrals by establishing new necessary conditions based on resonances and lowest order terms.

Findings

Provides necessary conditions for generalized rational first integrals

Extends classical results like Poincaré and Furta's work

Highlights the importance of lowest order rational homogeneous terms

Abstract

In this paper we mainly study the necessary conditions for the existence of functionally independent generalized rational first integrals of ordinary differential systems via the resonances. The main results extend some of the previous related ones, for instance the classical Poincar\'e's one \cite{Po}, the Furta's one, part of Chen's ones, and the Shi's one. The key point in the proof of our main results is that functionally independence of generalized rational functions implies the functionally independence of their lowest order rational homogeneous terms.