Planar polynomial foliations

Abstract:

Let $P(x,y)$ and $Q(x,y)$ be two real polynomials of degree $\leqslant n$ with no common real zeros. The solution curves of the vector field $\dot x = P(x,y),\dot y = Q(x,y)$ give a foliation of the plane. The leaf space $\mathcal {L}$ of this foliation may not be a hausdorff space: there may be leaves 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 $n \geqslant 4$, of a planar polynomial foliation of degree n whose leaf space contains $2n - 4$ such leaves.