Formal inverse integrating factors and the nilpotent center problem
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We are interested in deepening knowledge of methods based on formal power series applied to the nilpotent center problem of planar local analytic monodromic vector fields X. As formal integrability is not enough to characterize such a centers we use a more general object, namely, formal inverse integrating
factors V of X. Although by the existence of V is not possible to describe all nilpotent centers strata, we simplify, improve and also extend previous results on the relationship between these concepts. We use in the performed analysis the so-called Andreev number n N with n > 2 associated to X which is invariant under orbital conjugacy of X. Besides the leading terms in the (1,n)-quasihomogeneous expansions that V can have we also prove the following: (i) If n is even and there exists V then X has a center; (iii) If the existence of V characterizes all the centers; (iii) If there is a V with minimum ``vanishing multiplicity' at the singularity then, generically, X has a center.