{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T03:17:07Z","timestamp":1784258227918,"version":"3.55.0"},"reference-count":23,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2016,1]]},"abstract":"<jats:p> We are interested in deepening the knowledge of methods based on formal power series applied to the nilpotent center problem of planar local analytic monodromic vector fields [Formula: see text]. As formal integrability is not enough to characterize such a center we use a more general object, namely, formal inverse integrating factors [Formula: see text] of [Formula: see text]. Although by the existence of [Formula: see text] it 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 [Formula: see text] with [Formula: see text] associated to [Formula: see text] which is invariant under orbital equivalency of [Formula: see text]. Besides the leading terms in the [Formula: see text]-quasihomogeneous expansions that [Formula: see text] can have, we also prove the following: (i) If [Formula: see text] is even and there exists [Formula: see text] then [Formula: see text] has a center; (ii) if [Formula: see text], the existence of [Formula: see text] characterizes all the centers; (iii) if there is a [Formula: see text] with minimum \u201cvanishing multiplicity\u201d at the singularity then, generically, [Formula: see text] has a center. <\/jats:p>","DOI":"10.1142\/s0218127416500152","type":"journal-article","created":{"date-parts":[[2016,2,19]],"date-time":"2016-02-19T06:18:57Z","timestamp":1455862737000},"page":"1650015","source":"Crossref","is-referenced-by-count":14,"title":["Formal Inverse Integrating Factor and the Nilpotent Center Problem"],"prefix":"10.1142","volume":"26","author":[{"given":"Isaac A.","family":"Garc\u00eda","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tica, Universitat de Lleida, Avda. 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