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On the number of zeros of Abelian integrals

A constructive solution of the infinitesimal Hilbert sixteenth problem

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Abstract

We prove that the number of limit cycles generated from nonsingular energy level ovals (periodic trajectories) in 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 infinitesimal 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 defined over ℚ (the Gauss-Manin connection) with a quasiunipotent monodromy group.

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  1. Weizmann Institute of Science, Rehovot, Israel

    Gal Binyamini, Dmitry Novikov & Sergei Yakovenko

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  1. Gal Binyamini
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  2. Dmitry Novikov
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  3. Sergei Yakovenko
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Correspondence to Sergei Yakovenko.

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To Yulij Sergeevich Ilyashenko, who discovered this problem 40 years ago, for his 65th birthday with gratitude and admiration.

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Binyamini, G., Novikov, D. & Yakovenko, S. On the number of zeros of Abelian integrals. Invent. math. 181, 227–289 (2010). https://doi.org/10.1007/s00222-010-0244-0

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