On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:209

Article

Zhao, YL

关键词: Abelian integrals;    Picard-Fuchs systems;   

DOI  :  10.1016/j.jde.2004.07.007

来源: Elsevier

【 摘 要 】

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e.. its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence. we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k + n, under polynomial perturbation of arbitrary degree. (C) 2004 Elsevier Inc. All rights reserved.

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