# zbMATH — the first resource for mathematics

The generic rational differential equation $$dw/dz=P_n(z,w)/Q_n(z,w)$$ on $$\mathbb{C}\mathbb{P}^2$$ carries no interesting transverse structure. (English) Zbl 1018.37025
Authors’ abstract: Let $$\Gamma$$ be a non-solvable pseudogroup of holomorphic transformations in one variable fixing zero. Then for any $$z_0$$ sufficiently near zero and outside some real analytic set containing zero and depending on $$\Gamma$$, for any germ of biholomorphism $$\varphi$$ defined at $$z_0$$ with $$\varphi(z_0) -z_0$$ sufficiently small, there exists a sequence $$\gamma_n \in \Gamma$$ which tends to $$\varphi$$ uniformly on some neighborhood of zero. If we let $$\Gamma$$ be the holonomy pseudogrop of a compact leaf, we find that holomorphic codimension-1 foliations with non-solvable holonomy admit no transverse geometric structure in addition to the conformal one. This applies, in particular, to the singular foliation induced on $$\mathbb{C}\mathbb{P}^2$$ by the differential equation $$dw/dz= P_n(z,w)/Q_n (z,w)$$ where $$P_n$$ and $$Q_n$$ are the generic polynomials of degree $$n$$, hence the title of this paper.

##### MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37F75 Dynamical aspects of holomorphic foliations and vector fields 32S65 Singularities of holomorphic vector fields and foliations 57R30 Foliations in differential topology; geometric theory
Full Text: