The Multiplicity Of Invariant Algebraic Curves

In this paper we introduce a concrete notion of multiplicity for invariant algebraiccurves in polynomial vector fields§analyze the character of generalized invariant al-gebraic curves based on reducible invariant algebraic curves, and expand the applica-tion of multiplicity. We prove that the reducible non-degenerate generalized invariantalgebraic curves can be factorized into the product of irreducible non-degenerate gen-eralized invariant algebraic curves if and only if there is an irreducible non-degenerategeneralized invariant algebraic curve. We concentrate on the infinitesimal multiplicityand analyze some properties of the integrable multiplicity.