Multiplicity of Invariant Algebraic Curves and Darboux Integrability

TL;DR

This paper introduces four new types of multiplicity for invariant algebraic curves in polynomial vector fields, explores their relationships, and enhances Darboux integrability theory by incorporating these concepts.

Contribution

It defines novel multiplicity notions for invariant algebraic curves and improves Darboux integrability analysis through these new concepts.

Findings

Four different multiplicity types are defined and related.

Enhanced Darboux integrability criteria are developed.

Insights into singularities and the line at infinity are provided.

Abstract

We define four different kinds of multiplicity of an invariant algebraic curve for a given polynomial vector field and investigate their relationships. After taking a closer look at the singularities and at the line of infinity, we improve the Darboux theory of integrability using these new notions of multiplicity.