This article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields. It introduces the notion of limit periodic sets and shows that the Hilbert's 16th problem can be reduced to the existence of a uniform bound on its cyclicity. It also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the Noetherian property of the Bautin ideal.