Robust Satisfiability of Systems of Equations

We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f:K → Rn on a finite simplicial complex K and α>0, it holds that each function g:K → Rn such that ║g−f║∞ ≤ α, 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 simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.