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arXiv · 0808.2952

On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem

Gal Binyamini·Dmitry Novikov·Sergei Yakovenko
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Abstract

We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.

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Gal Binyamini, Dmitry Novikov, Sergei Yakovenko. 2009-10-06. On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem. https://doi.org/10.1007/s00222-010-0244-0

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