The method of Puiseux series and invariant algebraic curves

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Liénard dynamical systems [Formula: see text], [Formula: see text] with [Formula: see text]. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Liénard dynamical systems with [Formula: see text] can have at most two distinct irreducible invariant algebraic curves simultaneously and, consequently, are not integrable with a rational first integral.

RATIONAL FIRST INTEGRALS FOR POLYNOMIAL VECTOR FIELDS ON ALGEBRAIC HYPERSURFACES OF ℝn+1

Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in ℝn of degree m is at least [Formula: see text], then the vector field has a rational first integral. Llibre and Zhang used only Linear Algebra to provide a shorter and easier proof of the result given by Jouanolou. We use ideas of Llibre and Zhang to extend the Jouanolou result to polynomial vector fields defined on algebraic regular hypersurfaces of ℝn+1, this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of ℝn+1.