Robust satisfiability of systems of equations

article Robust satisfiability of systems of equations published yes Peter Franek author Marek Krcál author 33E21118-F248-11E8-B48F-1D18A9856A87 UlWa department HeEd department 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 dimK ≤ 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. ACM2015 eng Journal of the ACM 1402.0858 00036120050000110.1145/2751524 624 Franek P, Krcál M. 2015. Robust satisfiability of systems of equations. Journal of the ACM. 62(4), 26. P. Franek and M. Krcál, “Robust satisfiability of systems of equations,” Journal of the ACM, vol. 62, no. 4. ACM, 2015. Franek P, Krcál M. Robust satisfiability of systems of equations. Journal of the ACM. 2015;62(4). doi:<a href="https://doi.org/10.1145/2751524">10.1145/2751524</a> P. Franek, M. Krcál, Journal of the ACM 62 (2015). Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.” Journal of the ACM, vol. 62, no. 4, 26, ACM, 2015, doi:<a href="https://doi.org/10.1145/2751524">10.1145/2751524</a>. Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.” Journal of the ACM. ACM, 2015. <a href="https://doi.org/10.1145/2751524">https://doi.org/10.1145/2751524</a>. Franek, P., & Krcál, M. (2015). Robust satisfiability of systems of equations. Journal of the ACM. ACM. <a href="https://doi.org/10.1145/2751524">https://doi.org/10.1145/2751524</a> 16822018-12-11T11:53:27Z2025-09-23T10:38:46Z