Direct methods for primary decomposition

Eisenbud, David; Huneke, Craig; Vasconcelos, Wolmer V.

doi:10.1007/bf01231331

Cited by 182 publications

(119 citation statements)

References 26 publications

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“…There is a large literature on the problem of finding the radical ideal √ I of an ideal I; see, e.g., [3], [8], [17], [18], [22]. In the zero-dimensional case the problem is considered to be well-solved, e.g., via the following method of Seidenberg [35]: √ I = I ∪ {q 1 , .…”

Section: Laas-cnrs Andmentioning

confidence: 99%

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

2007

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For an ideal I ⊆ R [x] given by a set of generators, a new semidefinite characterization of its real radical I(V R (I)) is presented, provided it is zero-dimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety V R (I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gröbner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.

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Section: Laas-cnrs Andmentioning

confidence: 99%

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

2007

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“…A basis of this ideal can be computed over any computable field using Gröbner bases [5, Section 4.4]. Then, the generators of the associated primes of the ideal can be computed over any computable field with the splitting algorithm [6]. If there is only one associated prime, the ideal is prime.…”

Section: Basic Definitionsmentioning

confidence: 99%

On the generalized Ritt problem as a computational problem

Golubitsky¹,

Kondratieva²,

Ovchinnikov³

2009

J Math Sci

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Abstract. The Ritt problem asks if there is an algorithm that tells whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing if a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both proposed representations of a radical differential ideal are independent of the given set of generators and can be made independent of the ranking.

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“…It is a direction application of [14]. This ideal division (I : J(I)) can be computed by classical algebraic techniques, such as Gröbner bases [10].…”

Section: The Polar Varietymentioning

confidence: 99%

Regularity Criteria for the Topology of Algebraic Curves and Surfaces

Alberti

Mourrain

2007

Mathematics of Surfaces XII

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Abstract. In this paper, we consider the problem of analysing the shape of an object defined by polynomial equations in a domain. We describe regularity criteria which allow us to determine the topology of the implicit object in a box from information on the boundary of this box. Such criteria are given for planar and space algebraic curves and for algebraic surfaces. These tests are used in subdivision methods in order to produce a polygonal approximation of the algebraic curves or surfaces, even if it contains singular points. We exploit the representation of polynomials in Bernstein basis to check these criteria and to compute the intersection of edges or facets of the box with these curves or surfaces. Our treatment of singularities exploits results from singularity theory such as an explicit Whitney stratification or the local conic structure around singularities. A few examples illustrate the behavior of the algorithms.

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Direct methods for primary decomposition

Cited by 182 publications

References 26 publications

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

On the generalized Ritt problem as a computational problem

Regularity Criteria for the Topology of Algebraic Curves and Surfaces

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