2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59846We 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.16 pages, LaTeXDynamical Systemstext