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Newton's method is a classical iterative approach for computing solutions to nonlinear equations. To overcome some of its drawbacks, one often considers a continuous adjoint form of Newton's method. This paper investigates the geometric structure of the trajectories produced by the continuous adjoint Newton's method for bivariate quadratics, a system of two quadratic polynomials in two variables, via eigenanalysis at its equilibrium points. The main ideas are illustrated using plots generated by a Maple program. 

Ramification points arise from singularities along solution paths of a homotopy. This paper considers ramification points of homotopies, elucidating the total number of ramification points and providing general theory regarding the properties of the set of ramification points over the same branch point. The general approach utilized in this paper is to view homotopies as lines in the parameter spaces of families of polynomial systems on a projective manifold. With this approach, the number of singularities of systems parameterized by pencils is computed under broad conditions. General conditions are given for when the singularities of the systems parameterized by a line in a space of polynomial systems have multiplicity two. General conditions are also given for there to be at most one singularity in the solution set of any system parameterized by such a line. Several examples are included to demonstrate the theoretical results.