Transversals of Complex Polynomial Vector Fields

Vector fields in the complex plane are defined by assigning the vector determined by the value P(z) to each point z in the complex plane, where P is a polynomial of one complex variable. We consider special families of so-called rotated vector fields that are determined by a polynomial multiplied by rotational constants. Transversals are a certain class of curves for such a family of vector fields that represent the bifurcation states for this family of vector fields. More specifically, transversals are curves that coincide with a homoclinic separatrix for some rotation of the vector field. Given a concrete polynomial, it seems to take quite a bit of work to prove that it is generic, i.e. structurally stable. This has been done for a special class of degree d polynomial vector fields having simple equilibrium points at the d roots of unity, d odd. In proving that such vector fields are generic, an important step was proving that the transversals possessed a certain characteristic. Understanding transversals might be the key to proving other polynomial vector fields are generic, and they are important in understanding bifurcations of polynomial vector fields in general. We consider two important examples of rotated families to argue this. There will be discussed several open questions concerning the number of transversals that can appear for a certain degree d of a polynomial vector field, and furthermore how transversals are analyzed with respect to bifurcations around multiple equilibrium points.