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Algebraic solutions of polynomial differential equations and foliations in dimension two. (English) Zbl 0677.58036
Holomorphic dynamics, Proc. 2nd Int. Colloq. Dym. Syst., Mexico City/Mex. 1986, Lect. Notes Math. 1345, 192-232 (1988).
[For the entire collection see Zbl 0653.00010.]
Let $${\mathcal F}$$ be a singular holomorphic foliation of $${\mathbb{C}}P(2)$$ by complex curves which in affine coordinates are defined by an equation $$P(x,y)dx+Q(x,y)dy=0$$ where P, Q are polynomials without common factors. For an algebraic leaf S of $${\mathcal F}$$ and a singularity $$p\in S$$ of $${\mathcal F}$$ the branches $$B_ 1,...,B_ k$$ of S at p are defined. (If p is not a singularity of S then there is only one leaf.) Generalizing a concept of C. Camacho and P. Sad [Ann. Math., II. Ser. 115, 579-595 (1982; Zbl 0503.32007)] for each branch $$B_ i$$ an index is defined. The main result shows that the sum C(S,$${\mathcal F})$$ over the indices of all branches of all singularities of $${\mathcal F}$$ belonging to S equals $$3\cdot dg(S)-\chi (S)+\sum \mu (B)$$, where $$\sum \mu (B)$$ is the sum over the Milnor numbers of all these branches B, dg(S) is the degree of S, and $$\chi$$ (S) is the intrinsic Euler characteristic of S (i.e. the Euler characteristic of the Riemannian surface obtained by blowing up the singularities of S). This generalizes a result of Camacho and Sad (loc. cit.) where algebraic leaves are considered which may contain singularities of $${\mathcal F}$$ but have no singularities as algebraic curves. For foliations $${\mathcal F}^ a$$degree dg($${\mathcal F})$$ is defined, and it is proved that the space $${\mathfrak X}_ n$$ of all $${\mathcal F}$$ with dg($${\mathcal F})=n$$ (n$$\geq 2)$$ contains an open and dense subset whose elements have no algebraic leaves. (That $${\mathfrak X}_ n$$ contains a generic set with this property has been proved by J. P. Jouanolou [“Equations de Pfaff algébriques”, Lect. Notes Math. 708 (1979; Zbl 0477.58002)].) There are hints for analogous results in the real case.
Reviewer: H.G.Bothe

##### MSC:
 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 32C25 Analytic subsets and submanifolds