A system of Ordinary Differential Equations on the plane is said to have a center at one of its singular points if all its trajectories around this point are closed. It is a classical problem to give explicit necessary and sufficient conditions for a system to have a center. Recently this problem has been related to some problems in analysis and algebra, in particular, to the vanishing problem of some moment-like expressions and to the composition factorization of analytic functions. We present some developments in this direction showing, in particular, how the analytic structure of the moments and the composition algebra enter directly the Algebraic Geometry of the "Center equations".