A certified numerical algorithm for the topology of resultant and discriminant curves. (May 2017)

• Record Type: Journal Article
• Authors: Imbach, Rémi; Moroz, Guillaume; Pouget, Marc
• Is Part Of: Journal of symbolic computation. Volume 80:Part 2(2017)
• Page Start: 285
• Page End: 306
• Publication Date: 2017-05
• Subjects: Topology of algebraic curves -- Resultant -- Discriminant -- Subresultant -- Numerical algorithm -- Singularities -- Interval arithmetic -- Node and cusp singularities; 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.2016.03.011
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Let C be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces P ( x, y, z ) = Q ( x, y, z ) = 0 (resp. P ( x, y, z ) = ∂ P ∂ z ( x, y, z ) = 0 ), and generically its singularities are nodes (resp. nodes and ordinary cusps). State-of-the-art numerical algorithms compute the topology of smooth curves but usually fail to certify the topology of singular ones. The main challenge is to find practical numerical criteria that guarantee the existence and the uniqueness of a singularity inside a given box B, while ensuring that B does not contain any closed loop of C . We solve this problem by first providing a square deflation system, based on subresultants, that can be used to certify numerically whether B contains a unique singularity p or not. Then we introduce a numeric adaptive separation criterion based on interval arithmetic to ensure that the topology of C in B is homeomorphic to the local topology at p . Our algorithms are implemented and experiments show their efficiency compared to state-of-the-art symbolic or homotopic methods.