Stability condition for nilpotent singularities by its complex separatrices

This work is focused in the center problem for nilpotent singularities of differential systems in the plane. Although there are involved methods to approach the center problem in this work, we present a new stability criteria based on the existence of the complex separatrices passing through the singular point, and independent of the normal form. The algebraic method needs only the computation of such complex separatrices up to certain order, which can be done with a computer algebra package.