Implicitization of rational hypersurfaces via linear syzygies: A practical overview

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 th...

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Bibliographic Details
Published in:Journal of symbolic computation Vol. 74; pp. 493 - 512
Main Authors: Botbol, Nicolás, Dickenstein, Alicia
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.05.2016
Subjects:
Implicitization
Matrix representation
Rational surface
Sparse polynomial
Syzygy
Implicitization
Sparse polynomial
Syzygy
Matrix representation
Rational surface
ISSN:0747-7171, 1095-855X
Online Access:Get full text
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Summary: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.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2015.09.001