Implicitization of hypersurfaces

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...

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Bibliographic Details
Published in:Journal of symbolic computation Vol. 81; pp. 20 - 40
Main Authors: Abbott, John, Bigatti, Anna Maria, Robbiano, Lorenzo
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.07.2017
Subjects:
Hypersurface
Implicitization
68W30
Hypersurface
Implicitization
13-04
14Q10
13P10
13P25
ISSN:0747-7171, 1095-855X
Online Access:Get full text
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Summary: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.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2016.11.002