2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63999In 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.20 pagesAlgebraic GeometryComplex Variables32L30; 14D05text