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.

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