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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Planar polynomial foliations


Authors: Stephen Schecter and Michael F. Singer
Journal: Proc. Amer. Math. Soc. 79 (1980), 649-656
MSC: Primary 58F18; Secondary 57R30
Addendum: Proc. Amer. Math. Soc. 83 (1981), 220.
MathSciNet review: 572321
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0572321-0
PII: S 0002-9939(1980)0572321-0
Keywords: Planar polynomial vector field, foliation, inseparable leaves
Article copyright: © Copyright 1980 American Mathematical Society