Inventiones mathematicae

, Volume 181, Issue 2, pp 227-289

First online:

On the number of zeros of Abelian integrals

A constructive solution of the infinitesimal Hilbert sixteenth problem
  • Gal BinyaminiAffiliated withWeizmann Institute of Science
  • , Dmitry NovikovAffiliated withWeizmann Institute of Science
  • , Sergei YakovenkoAffiliated withWeizmann Institute of Science Email author 

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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.

Mathematics Subject Classification (2000)

34C07 34C08 34M10 34M60 14Q20 32S40