Upper bounds for the number of orbital topological types of planar polynomial vector fields modulo limit cycles

The paper deals with planar polynomial vector fields. We aim to estimate the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilberts 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. This paper is the continuation of the authors paper in [Mosc. Math. J. 1 (2001), no. 4] where the lower bound of the form 2^{cn^2} has been obtained. Here we obtain the upper bound of the same form. We also associate an equipped planar graph to every planar polynomial vector field, this graph is a complete invariant for orbital topological classification of such fields.