  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.
