Resultant over the residual of a complete intersection

Busé, Laurent; Elkadi, Mohamed; Mourrain, Bernard

doi:10.1016/s0022-4049(00)00144-4

Cited by 23 publications

(22 citation statements)

References 13 publications

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“…If we take an embedding of X corresponding to a linear series |dL − e 1 E i |, then the vanishing of the Chow form is the condition for 3 forms of degree d that vanish on E to vanish at a further common point. (This has also been called the "residual resultant", studied in the case of complete intersections in [Busé et al 2001].) We are able to find rank 2 Ulrich sheaves, corresponding to Pfaffian Bézout formulas for the resultant, if the ideal of the set of base points E is generated in degree < d.…”

Section: Surfacesmentioning

confidence: 99%

Resultants and Chow forms via exterior syzygies

Eisenbud

Schreyer²,

Weyman

2003

J. Amer. Math. Soc.

223

242

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Let W be a vector space of dimension n + 1 over a field K. The Chow divisor of a k-dimensional variety X in P n = P(W ) is the hypersurface, in the Grassmannian G k+1 of planes of codimension k + 1 in P n , whose points (over the algebraic closure of K) are the planes that meet X. The Chow form of X is the defining equation of the Chow divisor. For example, the resultant of k + 1 forms of degree e in k + 1 variables is the Chow form of P k embedded by the e-th Veronese mapping in P n with n = k+e kIn this paper we will give a new expression for the Chow divisor and apply it to give explicit formulas in many new cases. Starting with a sheaf F on P n , we use exterior algebra methods to define a canonical and effectively computable Chow complex of F on each Grassmannian of planes in P n . If F has k-dimensional support, we show that the Chow form of F is the determinant of the Chow complex of F on the Grassmannian of planes of codimension k + 1. The Beilinson monad of F [Beilinson 1978] is the Chow complex of F on the Grassmannian of 0-planes (that is, on P n itself.) In particular, we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks-Mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4, 6 or 8 in five variables. The easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables, the Chow form of the Veronese surface in P 5 . Using the tangent bundle of P 2 , conclude that it can be written in "Bézout form" (described below) as the Pfaffian of the matrix

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Section: Surfacesmentioning

confidence: 99%

Resultants and Chow forms via exterior syzygies

Eisenbud

Schreyer²,

Weyman

2003

J. Amer. Math. Soc.

223

242

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“…In order to deal with this base-point and get a non-identically zero resultant, we consider the blow-up π P : aP2 → a P 2 of the weighted projective space along the point P whose definition ideal is I P := (x, y), with exceptional divisor D. Following techniques developed in [2,6] (see also [7] for a quick overview), we obtain the so-called residual resultant, and denoted by Res aP2 (P h 1 , P h 2 , P h 3 ). It provides a necessary and sufficient condition for the zero locus in a P 2 of the system P h 1 (x, y, z) = P h 2 (x, y, z) = P h 3 (x, y, z) = 0 to be, scheme-theoretically, strictly bigger than the point P .…”

Section: The Residual Casementioning

confidence: 99%

“…It provides a necessary and sufficient condition for the zero locus in a P 2 of the system P h 1 (x, y, z) = P h 2 (x, y, z) = P h 3 (x, y, z) = 0 to be, scheme-theoretically, strictly bigger than the point P . For more details, we refer the reader to [2,7,6].…”

Section: The Residual Casementioning

confidence: 99%

Intersection and self-intersection of surfaces by means of Bezoutian matrices

Busé

Elkadi

Galligo

2008

Computer Aided Geometric Design

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The computation of intersection and self-intersection loci of parameterized surfaces is an important task in Computer Aided Geometric Design. We address these problems via four resultants with separated variables; two of them are specializations of general multivariate resultants and the two others are specializations of determinantal resultants. We give a rigorous study in these four cases and provide new formulas in terms of Bezoutian matrix.

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“…A more explicit proof is given in [15]. This quantity R 0 is also related to a residual resultant [2] and the subresultant [4].…”

Section: When Does This Work?mentioning

confidence: 99%

Accurate solution of polynomial equations using Macaulay resultant matrices

Jónsson

Vavasis

2004

Math. Comp.

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Abstract. We propose an algorithm for solving two polynomial equations in two variables. Our algorithm is based on the Macaulay resultant approach combined with new techniques, including randomization, to make the algorithm accurate in the presence of roundoff error. The ultimate computation is the solution of a generalized eigenvalue problem via the QZ method. We analyze the error due to roundoff of the method, showing that with high probability the roots are computed accurately, assuming that the input data (that is, the two polynomials) are well conditioned. Our analysis requires a novel combination of algebraic and numerical techniques.

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Resultant over the residual of a complete intersection

Cited by 23 publications

References 13 publications

Resultants and Chow forms via exterior syzygies

Resultants and Chow forms via exterior syzygies

Intersection and self-intersection of surfaces by means of Bezoutian matrices

Accurate solution of polynomial equations using Macaulay resultant matrices

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