Moderne Algebraische Geometrie

Gröbner, Wolfgang

doi:10.1007/978-3-7091-5740-4

Cited by 127 publications

(27 citation statements)

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“…The Proposition does yield new proofs for Serre's and Grobner's results on lengths and multiplicities, see [14], Chapter V, [4], 144.12 and [2]. COROLLARY …”

Section: And Equality Holds If and Only If A(x; C) And A(y; C) Are Comentioning

confidence: 98%

A criterion for intersection multiplicity one

Achilles

Huneke

Vogel

1985

Nagoya Mathematical Journal

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“…The Proposition does yield new proofs for Serre's and Grobner's results on lengths and multiplicities, see [14], Chapter V, [4], 144.12 and [2]. COROLLARY …”

Section: And Equality Holds If and Only If A(x; C) And A(y; C) Are Comentioning

confidence: 98%

A criterion for intersection multiplicity one

Achilles

Huneke

Vogel

1985

Nagoya Mathematical Journal

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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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“…the H-grading gives Macaulay's inverse systems, see [13], the oldest normal form interpolation space, while the least interpolant is obtained from using the H-grading together with the inner product…”

Section: Proof Since Both G and G Are γ-Bases For I We Conclude Fromentioning

confidence: 99%