Existence Verification for Higher Degree Singular Zeros of Complex Nonlinear Systems∗

Abstract. It is known that, in general, no computational techniques can verify the existence of a singular solution of the nonlinear system of n equations in n variables within a given region x of n-space. However, computational verification that a given number of true solutions exist within a region in complex space containing x is possible. That can be done by computation of the topological degree. In a previous paper, we presented theory and algorithms for the simplest case, when the rank-defect of the Jacobi matrix at the solution is one and the topological index is 2. Here, we will generalize that result to arbitrary topological index d ≥ 2: We present theory, algorithms, and experimental results. We also present a heuristic for determining the degree, obtaining a value that we can subsequently verify with our algorithms.