Center conditions: pull-back of differential equations

The space of polynomial dierential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull-back dierential equations form an irreducible component of such a space. The method used in this article is inspired by Ilyashenko and Movasati's method. The main concepts are the Picard-Lefschetz theory of a polynomial in two variables with complex coffcients, the Dynkin diagram of the polynomial and the iterated integral.