Characterization of a monodromic singular point of a planar vector field. A. Algaba, C. Garc´ ıa, M. Reyes. Department of Mathematics. Faculty of Experimental Sciences. Avda. Tres de Marzo s/n, 21071 Huelva, Spain e-mail: colume@uhu.es Version: March 10, 2010 The Newton diagram and the lowest-degree quasi-homogeneous terms of an analytic planar vector field allow us to determine whether an isolated singular point of the vector field is monodromic or has a characteristic trajectory. 1. INTRODUCTION. We are interested in the behavior of the trajectories in a neighborhood of a singular point of the planar analytic differential system ˙ x = F(x), (1) and, in particular, in determining when a singular point (we can assume the origin to be the singular point) is surrounded by orbits of the system (monodromic singular point). Each trajectory by lying on a vicinity of a monodromic singular point is either a spiral or a circle. Moreover, from the finiteness theorem for the number of limit cycles, a monodromic point of an analytic planar vector field can be only either a focus or a center, see Il’yashenko [13]. So, the monodromy problem is a previous step to solve the center problem of a vector field which is one of the classical open problems in the qualitative theory of planar differential systems. If the differential matrix DF(0) is not identically null, the monodromy problem is completely solved. The problem when the eigenvalues of the matrix are imaginary, was solved by Poincar´ e [17] and when the matrix is nilpotent, by Andreev [7]. Finally, if DF(0) is identically null (in such a case, O is a degenerate singular point), the monodromy problem can be solved by using blow-up technique (developed by Dumortier [10]) which consists of performing a series of changes to desingularize the point. How- ever, its application for determining the monodromy of a singular point of a family of vector fields with parameters becomes rather complicated. Some works that use this technique in order to study the monodromy are 1