Planar polynomial foliations
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- by Stephen Schecter and Michael F. Singer PDF
- Proc. Amer. Math. Soc. 79 (1980), 649-656 Request permission
Addendum: Proc. Amer. Math. Soc. 83 (1981), 220.
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.References
- A. A. Andronov et al., Qualitative theory of second order dynamic systems, Wiley, New York, 1973.
- L. Markus, Topological types of polynomial differential equations, Trans. Amer. Math. Soc. 171 (1972), 157–178. MR 306634, DOI 10.1090/S0002-9947-1972-0306634-4
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 79 (1980), 649-656
- MSC: Primary 58F18; Secondary 57R30
- DOI: https://doi.org/10.1090/S0002-9939-1980-0572321-0
- MathSciNet review: 572321