Multiplicity of Invariant Algebraic Curves and Darboux Integrability
  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. 
Jaume Llibre and Jorge Vitorio Pereira