Lower bounds for the number of orbital topological types of planar polynomial vector fields “modulo limit cycles”

We consider planar polynomial vector fields. We aim to find the (asymptotic) upper and lower bounds for the number of orbital topological equivalence classes for the fields of degree $n$. An evident obstacle for this is the second part of Hilbert's 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. Both upper and lower bounds can be obtained for this type of equivalence. In this paper we use the Viro gluing method to obtain the lower bound $2^{cn^2}$, where $c>0$ is a constant.