Newton's method and the Computational Complexity of the Fundamental Theorem of Algebra

Several different uses of Newton's method in connection with the Fundamental Theorem of Algebra are pointed out. Theoretical subdivision schemes have been combined with the numerical Newton iteration to yield fast root-approximation methods together with a constructive proof of the fundamental theorem of algebra. The existence of the inverse near a simple zero may be used globally to convert topological methods like path-following via Newton's method to numerical schemes with probabilistic convergence. Finally, fast factoring methods which yield root-approximations are constructed using some algebraic Newton iteration for initial factor approximations.