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

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 2n - 1 for all n >= 4. This improves the known lower bounds for s(n), which are 2n - 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) <= 2n 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. (C) 2020 Elsevier Inc. All rights reserved.