Center conditions: rigidity of logarithmic differential equations
- 作者：Movasati ; Hossein
- 关键词：32L30-14D05 ; Holomorphic foliations ; Picard&ndash ; Lefschetz theory
- 刊名：Journal of Differential Equations
- 出版年：2004
- 期刊代码：74_00220396
- 类别：mt
- 出版时间：February 10, 2004
- 卷：197
- 期：1
- 页码：197-217
- 文件大小：331 K
摘要
In this paper, we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko's result on Hamiltonian differential equations. The main tools are Picard–Lefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A'Campo's theorem on calculating the Dynkin diagram of the polynomial, and the action of Gauss–Manin connection on the so-called Brieskorn lattice/Petrov module of the polynomial. We will also generalize J.P. Francoise recursion formula and (*) condition for a polynomial which is a product of lines in a general position. Some applications on the cyclicity of cycles and the Bautin ideals will be given.