Invariant hyperplanes and Darboux integrability for d-dimensional polynomial differential systems

AbstractFor a class of polynomial differential systems of degree (m1,…,md) in Rd which is open and dense in the set of all polynomial differential systems of degree (m1,…,md) in Rd, we study the maximal number of invariant hyperplanes. This is a well known problem in dimension d=2 (see for instance [1,12,16]). Furthermore, using the Darboux theory of integrability we analyse when can be possible to find a first integral of a polynomial vector field of degree (m1,…,md) in Rd by knowing the existence of a sufficient number of invariant hyperplanes