The present invention relates to a process for synthesis of polymer compositions with reduced catalyst content, wherein ethylenically unsaturated monomers containing less than 0.5 wt % of ethylenically unsaturated monomers with at least one carboxylic group, sulfonic acid group and/or at least one phosphonic acid group relative to the total weight of the ethylenically unsaturated monomers are polymerized by means of initiators containing a transferable group of atoms and of one or more catalysts comprising at least one transition metal in the presence of ligands which can form a coordination compound with the metal catalyst or catalysts and, after the polymerization, the catalyst contained in the polymer is at least partly separated. For this purpose, the polymer composition is filtered, after the polymerization, in the presence of at least one filter aid which contains at least 0.5 wt %, relative to the total weight of the filter aid, of repeating units that contain at least one carboxylic acid group, sulfonic acid group and/or at least one phosphonic acid group.