Center conditions: Rigidity of logarithmic differential equations

Details Center conditions: Rigidity of logarithmic differential equations Center conditions: Rigidity of logarithmic differential equations

2002-05-07

arxiv.org/abs/math/0205068v2

Journal of Differential equations, 197 (2004), 197-217

Mathematics

Algebraic Geometry

20 pages

Scientific paper

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.

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