New lower bounds for the maximal number of inseparable leaves of n onsingular polynomial foliations of the plane

Abstract Let F be a nonsingular polynomial differential system of degree n on the real plane and denote by s ( n ) the maximal number of inseparable leaves that such a system can have. In this paper we prove that s ( n ) is at least 2 n − 1 for all n ≥ 4 . This improves the known lower bounds for s ( n ) , which are 2 n − 4 if n ≥ 7 or n = 5 , and respectively 6 and 9 if n = 4 and n = 6 . Since it is also known that s ( n ) ≤ 2 n for all n ≥ 4 and that s ( 0 ) = s ( 1 ) = 0 and s ( 2 ) = s ( 3 ) = 3 , the problem of determining s ( n ) for all n is now almost solved: any improvement in lower or upper bounds will actually find the exact s ( n ) . Our lower bounds for s ( n ) are attained in the class of Hamiltonian systems.