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Separatrices divide the phase space of some holomorphic dynamical systemsinto separate basins of attraction or 'stability regions' for distinct fixedpoints. 'Bundling' (high density) and mutual 'repulsion' of trajectories areoften observed at separatrices in phase portraits, but their globalmathematical characterisation is a difficult problem. For 1-D complexpolynomial dynamical systems we prove the existence of a separatrix for eachcritical point at infinity via transformation to the Poincar\'e sphere. We showthat introduction of complex time allows a significantly extended view with thestudy of corresponding Riemann surface solutions, their topology, geometry andtheir bifurcations/ramifications related to separatrices. We build a bridge tothe Riemann $\xi$-function and present a polynomial approximation of its Newtonflow solution manifold with precision depending on the polynomial degree.

On Analytical and Topological Properties of Separatrices in 1-D Holomorphic Dynamical Systems and Complex-Time Newton Flows