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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.

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
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