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 \to Rn on a finite simplicial complex K and alpha>0, it holds that each function g:K \to Rn such that ||g-f|| <= alpha, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming 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.