M ay 2 00 2 Center conditions : Rigidity of logarithmic differential equations

Abstract

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 PicardLefschetz 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. Some applications on the cyclicity of cycles and the Bautin ideals will be given.

Cite this paper

@inproceedings{Movasati2002MA2, title={M ay 2 00 2 Center conditions : Rigidity of logarithmic differential equations}, author={Hossein Movasati}, year={2002} }