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