Invariant hypersurfaces for derivations in positive characteristic

Let A be an integral k-algebra of finite type over an algebraically closed field k of characteristic p>0. Given a collection image of k-derivations on A, that we interpret as algebraic vector fields on image, we study the group spanned by the hypersurfaces V(f) of X invariant under image modulo the rational first integrals of image. We prove that this group is always a finite dimensional image-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k-algebra B between Ap and A, we show that the kernel of the pull-back morphism image is a finite image-vector space. In particular, if A is a UFD, then the Picard group of B is finite.