Singular points and limit cycles of planar polynomial vector fields (Q1975644)

The authors apply the ``gluing'' method to construct planar polynomial systems of the form \[ x'=P(x,y),\quad y'=Q(x,y), \] with ``large'' numbers of limit cycles or singular points. For example, they show that there exists an absolute constant \(C\) with the following property: it is possible to construct a system with \(\text{deg }P=\text{deg }Q=d\) having at least \[ \tfrac{1}{2} \log_2 d-C\log_2\log_2 d \] limit cycles. They also show that, for any nonnegative integer numbers \(d,s_0,s_1,s_2,\) and \(s_3\) satisfying the conditions \[ 2s_0+s_1+s_2+s_3=d^2,\quad |s_1+s_2-s_3|\leq d, \] it is possible to construct a system with \(\text{deg }P=\text{deg }Q=d\) having \(2s_0\) imaginary singular points, \(s_1\) attractors, \(s_2\) repellers, and \(s_3\) saddles.