Robust Satisfiability of Systems of Equations

Abstract: We study the problem of \emph{robust satisfiability} of systems of nonlinear equations, namely, whether for a given continuous function $f:\,K\to\mathbb{R}^n$ on a~finite simplicial complex $K$ and $\alpha>0$, it holds that each function $g:\,K\to\mathbb{R}^n$ such that $|g-f|_\infty \leq \alpha$, 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 $\dim K \le 2n-3$. This is a substantial extension of previous computational applications of \emph{topological degree} and related concepts in numerical and interval analysis. Via a reverse reduction we prove that the problem is undecidable when $\dim K\ge 2n-2$, where the threshold comes from the \emph{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.