2002/05/31 by Hossein Movasati
Mathematics · #math.AG #math.CV #msc:32L30 #msc:14D05
published as Journal of Differential equations, 197 (2004), 197-217 · 20 pages
arxiv created 2004/07/05 · arxiv updated 2009/11/30
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. Some applications on the cyclicity of cycles and the Bautin ideals will be given.