Upper Bounds for the Number of Orbital Topological Types of Planar Polynomial Vector Fields “Modulo Limit Cycles”

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.