Given a polynomial differential equation in \(\mathbb C^2, \dot z=P(z, w), \dot
w=Q(z, w)\) with complex time, we study the fibers of the corresponding
foliation, i.e., the splitting of \(\mathbb C^2\) into (real two-dimensional)
trajectories of this vector field.
A typical polynomial foliation on \(\mathbb C^2\) has many properties different from those of a typical (polynomial) foliation on \(\mathbb R^2\). For example, all fibers are dense in \(\mathbb C^2\), and there are infinitely many limit cycles.
However, many questions are still open, for example:
* What are the topological types of the fibers of a typical polynomial foliation?
* Take a polynomial foliation such that all leaves of its restriction to some domain in \(\mathbb C^2\) are cylinders (this is called an identical cycle). Is it possible to destroy this picture by a small perturbation of polynomials P,Q?
The talk will start with a brief review of the current state of the art, then I will talk about some recent results by Nataliya Goncharuk and myself.