Implicitization of rational hypersurfaces via linear syzygies: A practical overview. (May 2016)

• Record Type: Journal Article
• Authors: Botbol, Nicolás; Dickenstein, Alicia
• Is Part Of: Journal of symbolic computation. Volume 74(2016)
• Page Start: 493
• Page End: 512
• Publication Date: 2016-05
• Subjects: Implicitization -- Rational surface -- Syzygy -- Matrix representation -- Sparse polynomial; 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.2015.09.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We unveil in concrete terms the general machinery of the syzygy-based algorithms for the implicitization of rational surfaces in terms of the monomials in the polynomials defining the parametrization, following and expanding our joint article with M. Dohm. These algebraic techniques, based on the theory of approximation complexes due to J. Herzog, A. Simis and W. Vasconcelos, were introduced for the implicitization problem by J.-P. Jouanolou, L. Busé, and M. Chardin. Their work was inspired by the practical method of moving curves, proposed by T. Sederberg and F. Chen, translated into the language of syzygies by D. Cox. Our aim is to express the theoretical results and resulting algorithms into very concrete terms, avoiding the use of the advanced homological commutative algebraic tools which are needed for their proofs.