Robust satisfiability of systems of equations

Abstract

We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f: K → R n on a finite simplicial complex K and α > 0, it holds that each function g: K→Rn such that \\g - f//∞ ≤ a, has a root in K. Via a reduction to the extension problem of maps into a sphere, we show that this problem is decidable if dim K ≤ 2n - 3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction we prove that the problem is undecidable when dim K ≥ 2n - 2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is piecewise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings. Copyright © 2014 by the Society for Industrial and Applied Mathematics.