Invariant hyperplanes and Darboux integrability of polynomial vector fields

Abstract

This paper is composed of two parts. In the first part, we provide an upper bound for the number of invariant hyperplanes of the polynomial vector fields in n variables. This result generalizes those given in Artés et al (1998 Pac. J. Math. 184 207–30) and Llibre and Rodríguez (2000 Bull. Sci. Math. 124 599–619). The second part gives an extension of the Darboux theory of integrability to polynomial vector fields on algebraic varieties.