Geodesic
Upper Bounds for the Number of Orbital Topological Types of Planar Polynomial Vector Fields ``Modulo Limit Cycles''
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear analytic differential equations, Tome 254 (2006), pp. 254-271

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The purpose of this paper is to find an upper bound for the number of orbital topological types of $n$th-degree polynomial planar fields. An obstacle to obtaining such a bound is related to the unsolved second part of Hilbert's 16th problem. This obstacle is avoided by introducing the notion of equivalence modulo limit cycles. Earlier, the author obtained a lower bound of the form $2^{cn^2}$. In the present paper, an upper bound of the same form but with a different constant is found. Moreover, for each planar polynomial vector field with finitely many singular points, a marked planar graph is constructed that represents a complete orbital topological invariant of this field. 
@article{TM_2006_254_a11,
     author = {R. M. Fedorov},
     title = {Upper {Bounds} for the {Number} of {Orbital} {Topological} {Types} of {Planar} {Polynomial} {Vector} {Fields} {``Modulo} {Limit} {Cycles''}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {254--271},
     publisher = {mathdoc},
     volume = {254},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2006_254_a11/}
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R. M. Fedorov. Upper Bounds for the Number of Orbital Topological Types of Planar Polynomial Vector Fields ``Modulo Limit Cycles''. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear analytic differential equations, Tome 254 (2006), pp. 254-271. http://geodesic.mathdoc.fr/item/TM_2006_254_a11/