Abstract

We study the family of polynomial vector fields $xi_P(z) = P(z) frac d{dz}$ where $P$ is a monic centered polynomial of degree $d geq 2$. Any such vector field has $2(d-1)$ separatrices, counted with multiplicity, and labeled according to their asymptotic approach to infinity. The separatrix graph $Gamma_P$ is defined as the closure in $hat{mathbb C}$ of the separatrices for $xi_P$. The connected components of its complement, $hat{mathbb C} setminus Gamma_P$, fall into different kinds of zones. The topological structure of $Gamma_P$, together with its labeling, is described combinatorially and assigns a combinatorial invariant to $xi_P$. Each zone of $hat{mathbb C} setminus Gamma_P$ is assigned an analytic invariant. The combinatorial invariant and the set of analytic invariants for $xi_P$ form a complete invariant. The structure theorem guaranties that given a combinatorial data set and an analytic data set, satisfying certain rules, there exists a unique monic centered polynomial $P$ such that the invariants of $xi_P$ coincide with the given data set. The proof of the structure theorem is joined work with Ph.D.-student Kealey Dias