Certification for polynomial systems via square subsystems. (March 2022)

• Record Type: Journal Article
• Authors: Duff, Timothy; Hein, Nickolas; Sottile, Frank
• Is Part Of: Journal of symbolic computation. Volume 109(2022)
• Page Start: 367
• Page End: 385
• Publication Date: 2022-03
• Subjects: 65G20 -- 65H10; Certified solutions -- Alpha theory -- Polynomial system -- Numerical algebraic geometry -- Newton-Okounkov bodies -- Schubert calculus; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.07.010
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which are solutions to the original overdetermined system. These approaches each use different additional information for this certification, such as liaison, Newton-Okounkov bodies, or intersection theory. They may be used to certify individual solutions, reject nonsolutions, or certify that we have found all solutions.