Cyclicity of period annuli and principalization of Bautin ideals

We prove that the maximal number of limit cycles which bifurcate from an open period annulus under a given multi-parameter analytic deformation of a given analytic vector field is the same as in an appropriate one-parameter analytic deformation of the field, provided that this cyclicity is finite. Along the same lines we give also a bound of the cyclicity of homoclinic saddle loops.