Planar polynomial foliations

Abstract

Let P ( x , y ) P(x,y) and Q ( x , y ) Q(x,y) be two real polynomials of degree ⩽ n \leqslant n with no common real zeros. The solution curves of the vector field x ˙ = P ( x , y ) , y ˙ = Q ( x , y ) \dot x = P(x,y),\dot y = Q(x,y) give a foliation of the plane. The leaf space L \mathcal {L} of this foliation may not be a hausdorff space: there may be leaves L, L ′ ∈ L L’ \in \mathcal {L} which cannot be separated by open sets. We show that the number of such leaves is at most 2n and construct an example, for each even n ⩾ 4 n \geqslant 4 , of a planar polynomial foliation of degree n whose leaf space contains 2 n − 4 2n - 4 such leaves.