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By work of Darboux, and later Jouanolou, a polynomial vector field on the complex projective plane has only algebraic solutions if and only if it has sufficiently many such solutions, or equivalently, if there are enough algebraic curves left invariant by the vector field. In light of this, to decide whether a polynomial vector field is algebraically integrable we propose to relate numerical invariants not only of curves in the plane, but also of varieties in higher dimensional projective spaces, to the degree of vector fields that are tangent to these varieties. Since the work of Poincare, such ´ bounds have been investigated in many papers with different techniques. We plan to employ methods mainly from commutative algebra, combining fairly recent advances on Castelnuovo-Mumford regularity, in particular the regularity of ideal powers, with more classical tools from singularity theory.