On the separatrix graph of a rational vector field on the Riemann sphere

Abstract

We consider the rational flow ${\xi }_R(z)=R(z)(d/dz)$ where $R$ is given by the quotient of two polynomials without common factors on the Riemann sphere. The separatrix graph ${\Gamma }_R$ is the boundary between trajectories with different properties. We characterize the properties of a planar directed graph to be homeomorphic to the separatrix graph of a rational vector field on the Riemann sphere.

25 pages, 14 figures