Minors of Bezout matrices, subresultants and the parameterization of the degree of the polynomial greatest common divisor

Abdeljaoued, Jounaïdi; Diaz–Toca, Gema M.; González-Vega, Laureano

doi:10.1080/00207160412331284178

Cited by 14 publications

(27 citation statements)

References 11 publications

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“…For this specific setting, with our implementations of the various alternatives, and with our benchmark data, we have observed that Berkowitz' method for determinant computation is faster than Gauss-Bareiss, despite the asymptotics; and that evaluating the Bézout determinant with Berkowitz is faster than using the Sylvester determinant or a subresultant remainder sequence. This is consistent with the results of a recent study by Abdeljaoued et al [1] for polynomials with higher degrees and more inner variables.…”

Section: Computing the Resultantsupporting

confidence: 94%

“…. , s k , intersection multiplicities will typically not have been computed for all 1 2 k(k − 1) pairs of segments. For 1 i < k let m i ∈ {1, 2, 3, .…”

Section: Reordering Segments Passing Through An Eventmentioning

confidence: 99%

“…We call f ∈ R[x] monic if its leading coefficient (f ) is 1. A multivariate polynomial can be seen recursively as a univariate polynomial in x n with coefficients in R[x 1 ] · · · [x n−1 ], or flat as a sum of terms a i 1 ...i n x i 1 1 · · · x i n n . Its total degree deg(f ) is max{i 1 + · · · + i n | a i 1 ...i n = 0}.…”

Section: Basicsmentioning

confidence: 99%

“…The recent article[1] explains how to obtain all subresultants from one execution of Berkowitz' method. While quite interesting in general, this would not gain a lot in our setting where m 3, k 1 4.…”

mentioning

confidence: 99%

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Exact, efficient, and complete arrangement computation for cubic curves

Eigenwillig

Kettner

Schömer

et al. 2006

Computational Geometry

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The Bentley-Ottmann sweep-line method can compute the arrangement of planar curves, provided a number of geometric primitives operating on the curves are available. We discuss the reduction of the primitives to the analysis of curves and curve pairs, and describe efficient realizations of these analyses for planar algebraic curves of degree three or less. We obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Special cases of cubic curves are conics as well as implicitized cubic splines and Bézier curves.The algorithm is complete in that it handles all possible degeneracies such as tangential intersections and singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement. The algorithm has been implemented in C++ as an EXACUS library called CUBIX.

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Section: Computing the Resultantsupporting

confidence: 94%

“…. , s k , intersection multiplicities will typically not have been computed for all 1 2 k(k − 1) pairs of segments. For 1 i < k let m i ∈ {1, 2, 3, .…”

Section: Reordering Segments Passing Through An Eventmentioning

confidence: 99%

Section: Basicsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Exact, efficient, and complete arrangement computation for cubic curves

Eigenwillig

Kettner

Schömer

et al. 2006

Computational Geometry

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“…When p is a polynomial with constant coefficients, these sequences can be computed efficiently by a subresultant algorithm in Lombardi, Roy & Safely el Din (2000). However, the algorithm is not so efficient when p has a lot of parameters (Abdeljaoued, Diaz-Toca & Gonzalez-Vega, 2004). …”

Section: Sturm-habicht Sequence and Related Sequencesmentioning

confidence: 99%

An Algorithm for Computing the Complete Root Classification of a Parametric Polynomial

Liang

Jeffrey

2006

Artificial Intelligence and Symbolic Computation

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An improved algorithm, together with its implementation, is presented for the automatic computation of the complete root classification of a real parametric polynomial. The algorithm offers improved efficiency and a new test for non-realizable conditions. The improvement lies in the direct use of 'sign lists', obtained from the discriminant sequence, rather than 'revised sign lists'. It is shown that the discriminant sequences, upon which the sign lists are based, are closely related both to Sturm-Habicht sequences and to subresultant sequences. Thus calculations based on any of these quantities are essentially equivalent.One particular application of complete root classifications is the determination of the conditions for the positive definiteness of a polynomial, and here the new algorithm is applied to a class of sparse polynomials. It is seen that the number of conditions for positive definiteness remains surprisingly small in these cases.

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