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

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.Comment: 25 pages, 14 figure

Classification of Complex Polynomial Vector Fields in One Complex Variable

A classification of the global structure of monic and centered one-variable complex polynomial vector fields is presented.Comment: 57 pages, 35 figures, submitted to the Journal of Difference Equations and Application