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Abelian integrals in holomorphic foliations. (English) Zbl 1055.37057

Consider the differential equation \[ dy/dx= P(x, y)/Q(x, y)\tag{\(*\)} \] in the real plane \(\mathbb{R}^2\), where \(P\) and \(Q\) are polynomials in \(x\) and \(y\). Let \(H(P, Q)\) denote the number of limit cycles of the above differential equation \((*)\) and denote by \(H_n\) the maximum number (maybe infinity ) of \(H(P, Q)\) for \(\deg P\leq n\), \(\deg Q\leq n\). The question of whether \(H_n\) is finite or not is considered as the Hilbert’s 16th problem. To solve this problem, E. M. Landis and I. G. Petrovskij [Am. Math. Soc., Transl., II. Ser. 14, 181–199 (1960; Zbl 0094.06304)] complexified the equation and considered \((*)\) in \(\mathbb{C}^2\) and, by using Abelian integrals, tried to find another set \(C(P, Q)\) of cycles in the solutions of \((*)\) whose cardinality is not less than the number of limit cycles in question. The author introduces an algebro-geometric approach to the Abelian integrals on an arbitrary two-dimensional compact complex manifold \(M\) instead of \(\mathbb{C}^2\). In fact, the author considers holomorphic foliations on \(M\) and shows, using Abelian integrals, some relationships between some meromorphic (called Melnikov) functions and the number of limit cycles in the Hilbert’s 16th problem. The author shows the importance of the Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory.

MSC:

37F75 Dynamical aspects of holomorphic foliations and vector fields
32S65 Singularities of holomorphic vector fields and foliations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations

Citations:

Zbl 0094.06304
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References:

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