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The theory of Khovanskii and the problem of Dulac. (Théorie de Hovanskii et problème de Dulac.) (French) Zbl 0769.58050
This is a paper of great interest. The XVIth Hilbert problem, still unsolved, generates the so called Dulac problem which consists in showing that for an analytic equation in the real plane \(\mathbb{R}^ 2\), \(\dot x = f(x)\), limit cycles cannot accumulate (their number is locally finite). The Dulac problem has been solved by Il’yashenko and also by Ecalle, Martinet, Moussu, Ramis, but the proofs are very complex and difficult. Four years ago Moussu had an idea of approaching this problem geometrically rather than analytically. Instead of studying the Poincaré map he started studying the solutions. Like that he met the ideas of Khovanskij.
Interestingly, the solutions of an analytic equation \(\dot x = f(x)\) in the plane, although not being subanalytic in most of the cases, behave surprisingly well. Moussu worked on this with Roche and soon they found out that some techniques developed by the Polish school (Łojasiewicz, the reviewer, Stasica, Wachta) for subanalytic sets can be generalized to separating varieties and combined with the Rolle theorem of Khovanskii. Together with the good use of transversality, it gives the following Theorem 2. Let \(X\) and \(M\) be semianalytic in \(N\), \(M\) relatively compact, and let \(\omega\) be a 1-form analytic in a neighbourhood of \(\overline{M}\) and let \(V_ 1,\dots,V_ q\) be separating surfaces of Khovanskii (which means \(M\setminus V_ i\) is a union of two open connected sets with common border in \(M\) and for each \(x\in V\) \(\text{Ker }\omega(x)\) is tangent to \(T_ xV\) in \(x\)), then the number of connected components of \(X\cap V_ 1 \cap \dots \cap V_ q\) is finite. This was a tool for proving the following
Theorem 1. Let \(f\) be the map of first return for a polycycle \(P\) of an analytic differential equation \(\omega = 0\), where \(\omega = a(x,y)dx + b(x,y)dy\) in \(\mathbb{R}^ 2\), defined on an open subset in \(\mathbb{R}^ 2\).
If \(P\) verifies the hypothesis: \(\omega\) has an analytic integrating factor at each vertex, then either \(f = \text{id}\) or 0 is a fixed isolated point of \(f\). (This excludes the accumulation of limit cycles of \(P\).)
The existence of an analytic integrating factor is, unfortunately, quite restrictive, but the results obtained while trying to prove Dulac’s conjecture this way are extremely interesting and led to the notion of Pfaffian varieties and their study (to be published).
Reviewer: Z.Denkowska

MSC:
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
32B20 Semi-analytic sets, subanalytic sets, and generalizations
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