Multiple solutions of nonlinear equations: Roots of polynomials

A search method is presented for obtaining multiple solutions of a system of nonlinear equations whose first equations do not necessarily define a unique space curve. In particular, the approach is used to find all the roots of a complex polynomial. Singularities on the space curve are analyzed and properly classified according to their high-order derivatives. Depending on the nature of singularities, the rules for a sign change in the algorithm are determined so that the root-finding procedure can be continued.