{"date_updated":"2025-09-23T10:38:46Z","quality_controlled":"1","department":[{"_id":"UlWa"},{"_id":"HeEd"}],"issue":"4","type":"journal_article","arxiv":1,"external_id":{"arxiv":["1402.0858"],"isi":["000361200500001"]},"title":"Robust satisfiability of systems of equations","article_number":"26","language":[{"iso":"eng"}],"month":"08","publication":"Journal of the ACM","isi":1,"volume":62,"oa_version":"Preprint","year":"2015","abstract":[{"text":"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 &gt; 0, it holds that each function g: K → ℝn such that ||g - f || ∞ &lt; 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 &gt; 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.","lang":"eng"}],"_id":"1682","scopus_import":"1","article_processing_charge":"No","day":"01","publist_id":"5466","corr_author":"1","date_published":"2015-08-01T00:00:00Z","status":"public","oa":1,"citation":{"chicago":"Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.” <i>Journal of the ACM</i>. ACM, 2015. <a href=\"https://doi.org/10.1145/2751524\">https://doi.org/10.1145/2751524</a>.","ieee":"P. Franek and M. Krcál, “Robust satisfiability of systems of equations,” <i>Journal of the ACM</i>, vol. 62, no. 4. ACM, 2015.","mla":"Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.” <i>Journal of the ACM</i>, vol. 62, no. 4, 26, ACM, 2015, doi:<a href=\"https://doi.org/10.1145/2751524\">10.1145/2751524</a>.","ista":"Franek P, Krcál M. 2015. Robust satisfiability of systems of equations. Journal of the ACM. 62(4), 26.","short":"P. Franek, M. Krcál, Journal of the ACM 62 (2015).","apa":"Franek, P., &#38; Krcál, M. (2015). Robust satisfiability of systems of equations. <i>Journal of the ACM</i>. ACM. <a href=\"https://doi.org/10.1145/2751524\">https://doi.org/10.1145/2751524</a>","ama":"Franek P, Krcál M. Robust satisfiability of systems of equations. <i>Journal of the ACM</i>. 2015;62(4). doi:<a href=\"https://doi.org/10.1145/2751524\">10.1145/2751524</a>"},"publication_status":"published","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1402.0858"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.1145/2751524","date_created":"2018-12-11T11:53:27Z","publisher":"ACM","author":[{"last_name":"Franek","full_name":"Franek, Peter","first_name":"Peter"},{"first_name":"Marek","id":"33E21118-F248-11E8-B48F-1D18A9856A87","full_name":"Krcál, Marek","last_name":"Krcál"}],"intvolume":"        62"}