Rational first integrals for polynomial vector fields on algebraic hypersurfaces of ℝ ⁿ⁺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 ℝ n of degree m is at least {n+m-1 \choose n}+n, 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 ℝ n+1 , this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of ℝ n+1 . © 2012 World Scientific Publishing Company.