New lower bounds for the maximal number of inseparable leaves of nonsingular 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 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.