Sign Methods for Enumerating Solutions of Nonlinear Algebraic Systems

We implement the concept of topological degree to isolate and compute all zeros of systems of nonlinear algebraic equations when the only computable information required is the algebraic signs of the components of the function. The basic theorems of Kronecker–Picard theory relate the number of roots to the topological degree. They are combined with grid methods in order to compute the topological degree by using the minimum possible information, namely the sign of a function at some value. Recent fast methods, which work over fixed precision, are applied to determine the sign of algebraic systems. Keywords—Polynomial system solving, zero isolation, bisection, characteristic polytope, topological degree, sign determination.