Isolated points, duality and residues

Mourrain, Bernard

doi:10.1016/s0022-4049(97)00023-6

Cited by 69 publications

(88 citation statements)

References 9 publications

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“…. , f n+1 ) are connected to these matrices, including duality, algebraic residues (Scheja and Storch, 1975;Kunz, 1986;Elkadi and Mourrain, 1996), real root counting (Becker et al, 1996), multiplicities (Mourrain, 1996b), . .…”

Section: Comparison Between Different Matricesmentioning

confidence: 99%

Matrices in Elimination Theory

Emiris

Mourrain

1999

Journal of Symbolic Computation

105

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The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay's matrix and further conjecture the generalization of Macaulay's exact rational expression for the resultant polynomial to the toric case. A new theorem proves that the maximal minor of a Bézout matrix is a non-trivial multiple of the resultant. We discuss applications to constructing monomial bases of quotient rings and multiplication maps, as well as to system solving by linear algebra operations. Lastly, degeneracy issues, a major preoccupation in practice, are examined. Throughout the presentation, examples are used for illustration and open questions are stated in order to point the way to further research.

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Section: Comparison Between Different Matricesmentioning

confidence: 99%

Matrices in Elimination Theory

Emiris

Mourrain

1999

Journal of Symbolic Computation

105

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“…The multiplicity structure of a singular solution has been studied extensively in [2,21,22,24,7,9,8,26,25,13]. Various methods have also been proposed for computing the singular solutions to high accuracy [5,32,31,18,19,17].…”

Section: Resultsmentioning

confidence: 99%

Computing the multiplicity structure from geometric involutive form

Zhi

2008

Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation

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“…Since a zero-dimensional system has only finitely many zeros, each zero must be isolated in the sense of Definition 2 so the content of these theorems is simply the classical result that dim Dx(F ) is identical to the intersection multiplicity (cf. [10,16,21]) along with more recent expositions by Emsalem [7], Mourrain [24] and Stetter [29].…”

Section: Where (·)mentioning

confidence: 99%

Multiple zeros of nonlinear systems

Dayton

Zeng

2011

Math. Comp.

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Abstract. As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity structure, proposes a depth-deflation method for accurate computation of multiple zeros, and introduces the basic algebraic theory of the multiplicity.Furthermore, this paper elaborates and proves some fundamental properties of the multiplicity, including local finiteness, consistency, perturbation invariance, and depth-deflatability.As a justification of this formulation, the multiplicity is proved to be consistent with the multiplicity defined in algebraic geometry for the special case of polynomial systems.The proposed algorithms can accurately compute the multiplicity and the multiple zeros using floating point arithmetic even if the nonlinear system is perturbed.

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Isolated points, duality and residues

Cited by 69 publications

References 9 publications

Matrices in Elimination Theory

Matrices in Elimination Theory

Computing the multiplicity structure from geometric involutive form

Multiple zeros of nonlinear systems

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