Higher Order Turning Points

摘要

When a real homotopy is used for solving a system of nonlinear equations of complex variables, especially polynomial systems, bifurcation at singular points is inevitable. In this paper, the notion of a kth order turning point, a singular point with k bifurcation branches, is introduced. It is shown that the tangent lines of those k solution paths at a kth order turning point lie in the same one-dimensional complex plane and equally divide this plane. This result generalizes the phenomena taking place at a quadratic turning point where tangeant lines of the solution paths, only two of them, are perpendicular to each other. Moreover, our result provides a condition to verify the true multiplicity of a solution when several homotopy paths converge to it.