On the number of zeros of Abelian integrals

• 成果类型: Article
• 署名作者: Binyamini, Gal; Novikov, Dmitry; Yakovenko, Sergei
• 署名单位: Weizmann Institute of Science
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-010-0244-0
• 发表日期: 2010
• 页码: 227-289
• 关键词: ordinary differential-equations; exponential estimate; finite cyclicity; bounds; real; perturbations; POLYNOMIALS; oscillation; unfoldings; systems

摘要

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 ae (the Gauss-Manin connection) with a quasiunipotent monodromy group.