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Complex techniques to study ODE. (Techniques complexes d’étude d’E.D.O.) (French. English summary) Zbl 1156.37005

This interesting paper shows how to study local real analytic foliations of the real plane \({\mathbb R}^2\) by using the associated complexified local holomorphic foliation of \({\mathbb C}^2\).
The first notion studied in this paper is that of companion: a companion of a real leaf \(L\subset{\mathbb R}^2\) is a (different) connected component of the intersection of the complexified leaf \(L^{\mathbb C}\subset{\mathbb C}^2\) with \({\mathbb R}^2\). If the first order jet of the foliation is hyperbolic, the authors show that either the generic leaf has no companion, or if the generic leaf has a companion then the foliation admits an analytic first integral (or, up to an analytic conjugation, a first integral of the form \(x_1^q/x_2^p\)). If the first order jet of the foliation is elliptic, then generic leaves have no companion. Finally, when the first order jet is a saddle-node or is identically zero, the authors present examples showing how more complicated behaviors can occur.
The second object studied here is the saturation in \({\mathbb C}^2\) of the plane \({\mathbb R}^2\) with respect to the complexified foliation. The authors first compute the saturation for some foliations in normal forms, showing that it is in general Pfaffian (i.e., a leaf of a real rank 3 foliation in \({\mathbb C}^2\)) but not necessarily analytic (in particular, not necessarily a hypersurface). Conversely, they show that if the first jet is hyperbolic, saddle-node or elliptic, and the saturation is Pfaffian, then the real foliation is either linearizable or analytically conjugated to its formal normal form. The proof depends on an interesting theorem proved by J. Ribón in the Appendix, saying that a holomorphic germ tangent to the identity in dimension 1 is holomorphically conjugated to its formal normal form if it preserves a real analytic foliation of \({\mathbb R}^2\). Finally, the authors prove that if the first jet of the foliation is of the form \(qx_1\,dx_2+px_2\,dx_1\) with \(p\in{\mathbb Z}^*\) and \(q\in{\mathbb N}^*\), and the saturation is an analytic hypersurface, then the foliation admits an analytic first integral if \(p>0\), and (up to an analytic conjugation) a first integral of the form \(x_1^px_2^q\) if \(p<0\).
Reviewer: Marco Abate (Pisa)

MSC:

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37E35 Flows on surfaces
37F75 Dynamical aspects of holomorphic foliations and vector fields
32S65 Singularities of holomorphic vector fields and foliations
34A26 Geometric methods in ordinary differential equations
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
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