Inventiones mathematicae

, Volume 181, Issue 2, pp 227–289

On the number of zeros of Abelian integrals

A constructive solution of the infinitesimal Hilbert sixteenth problem

DOI: 10.1007/s00222-010-0244-0

Cite this article as:
Binyamini, G., Novikov, D. & Yakovenko, S. Invent. math. (2010) 181: 227. doi:10.1007/s00222-010-0244-0


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)


Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Gal Binyamini
    • 1
  • Dmitry Novikov
    • 1
  • Sergei Yakovenko
    • 1
  1. 1.Weizmann Institute of ScienceRehovotIsrael