Three new algorithms for multivariate polynomial GCD
Journal of Symbolic Computation
Symbolic and numeric methods for exploiting structure in constructing resultant matrices
Journal of Symbolic Computation
Elimination theory in codimension 2
Journal of Symbolic Computation
Numerical Implicitization of Parametric Hypersurfaces with Linear Algebra
AISC '00 Revised Papers from the International Conference on Artificial Intelligence and Symbolic Computation
Implicitization of rational surfaces by means of polynomial interpolation
Computer Aided Geometric Design
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Computing Approximate GCDs in Ill-conditioned Cases
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
ApaTools: a software toolbox for approximate polynomial algebra
ACM Communications in Computer Algebra
Implicit polynomial support optimized for sparseness
ICCSA'03 Proceedings of the 2003 international conference on Computational science and its applications: PartIII
Approximate GCD of several univariate polynomials with small degree perturbations
Journal of Symbolic Computation
An output-sensitive algorithm for computing projections of resultant polytopes
Proceedings of the twenty-eighth annual symposium on Computational geometry
Implicitization of curves and surfaces using predicted support
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
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We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiple of the implicit equation; the latter can be obtained via multivariate polynomial GCD (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of a bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial.