On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree

In this contribution the isolation of real roots and the computation of the topological degree in two dimensions are considered and their complexity is analyzed. In particular, we apply Stenger's degree computational method by splitting properly the boundary of the given region to obtain a sequence of subintervals along the boundary that forms a sufficient refinement. To this end, we properly approximate the function using univariate polynomials. Then we isolate each one of the zeros of these polynomials on the boundary of the given region in various subintervals so that these subintervals form a sufficiently refined boundary.