Implicitization of hypersurfaces. (July 2017)

• Record Type: Journal Article
• Authors: Abbott, John; Bigatti, Anna Maria; Robbiano, Lorenzo
• Is Part Of: Journal of symbolic computation. Volume 81(2017)
• Page Start: 20
• Page End: 40
• Publication Date: 2017-07
• Subjects: 13P25 -- 13P10 -- 13-04 -- 14Q10 -- 68W30; Hypersurface -- Implicitization; 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.2016.11.002
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, "ElimTH", has as main step the computation of an elimination ideal via a truncated, homogeneous Gröbner basis. The other algorithm, "Direct", computes the implicitization directly using an approach inspired by the generalized Buchberger–Möller algorithm. Either may be used inside the third algorithm, "RatPar", to deal with parametrizations by rational functions. Finally we show how these algorithms can be used in a modular approach, algorithm "ModImplicit", for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.