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

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.

Final revision

References in corpus (5)

• Redundant Picard-Fuchs system for Abelian integrals
• Bounded decomposition in the Brieskorn lattice and Pfaffian Picard--Fuchs systems for Abelian integrals
• Oscillation of linear ordinary differential equations: on a theorem by A. Grigoriev
• Quasialgebraicity of Picard--Vessiot fields
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Cited by in corpus (5)

• The $16$th Hilbert problem on algebraic limit cycles
• Point counting for foliations over number fields
• A Uniform Version of the Petrov-Khovanskii Theorem
• On the reduction of the degree of linear differential operators
• Effective de Rham Cohomology - The General Case