Invariant hyperplanes and Darboux integrability for [formula omitted]-dimensional polynomial differential systems

Abstract

For a class of polynomial differential systems of degree (m 1,…,m d) in R d which is open and dense in the set of all polynomial differential systems of degree (m 1,…,m d) in R d , 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 (m 1,…,m d) in R d by knowing the existence of a sufficient number of invariant hyperplanes.