Can we approximate zeros of functions with nonzero topological degree?
The topological degree of a scalar continuous function f, defined on an interval [a,b] of the real-axis, decides whether we can or cannot solve the problem of finding an \(\epsilon\)-approximation \(x^*\), \(| x^*- \alpha (f)| \leq \epsilon\), to a zero \(\alpha\) (f) of f, since if we assume that \(f(a)f(a)<0\) then \(\deg (f,[a,b],0)=(sgn(f(b))- sgn(f(a)))/2\neq 0\) and, as the second author has proved [Numer. Math. 40, 111-117 (1982; Zbl 0492.65027)], the bisection method for finding such an approximation is optimal. If the degree is zero then, in general, there exists no algorithm using linear information of f to find \(x^*.\) The authors are able to show that the situation drastically changes when we add just one more dimension. They prove that, in general, it is impossible to find an \(\epsilon\)-approximation to a zero of a smooth bivariate function with nonzero topological degree.
scientific article; zbMATH DE number 4098625