The Calculation of Radical Ideals in Positive Characteristic

Kemper, Gregor

doi:10.1006/jsco.2002.0560

Cited by 18 publications

(13 citation statements)

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“…Here an efficient algorithm means that if N = dim A = O(g), then we have a Las Vegas algorithm with an expected complexity that is polynomial in g, where we need to measure complexity in terms of both field operations and factorizations of degree O(N ) polynomials in k [x]. As examples of algorithms for primary decomposition and the computation of radicals, we mention Chapter 8 of [BW93], as well as the articles [EG00], [Kem02], and [DGP99], and the articles cited in their bibliographies.…”

Section: Converting From Representation a To Representation Bmentioning

confidence: 99%

Asymptotically fast group operations on Jacobians of general curves

Khuri-Makdisi

2007

Math. Comp.

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Abstract. Let C be a curve of genus g over a field k. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of C. After a precomputation, which is done only once for the curve C, the algorithms use only linear algebra in vector spaces of dimension at most O(g log g), and so take O(g 3+ ) field operations in k, using Gaussian elimination. Using fast algorithms for the linear algebra, one can improve this time to O(g 2.376 ). This represents a significant improvement over the previous record of O(g 4 ) field operations (also after a precomputation) for general curves of genus g.

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Section: Converting From Representation a To Representation Bmentioning

confidence: 99%

Asymptotically fast group operations on Jacobians of general curves

Khuri-Makdisi

2007

Math. Comp.

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“…However, if k does not have characteristic 0, k(u) might not be perfect. In this case, more elaborated algorithms ( [14], [18]) can be used. We will restrict to the case of characteristic 0.…”

Section: Preliminariesmentioning

confidence: 99%

“…Our routine uses the subroutine for the reduction to the zero dimensional case that is already implemented in the library primdec [5] for the computation of the radical by Krick-Logar-Kemper algorithm. We compare the times obtained by our algorithm with the algorithms implemented in primdec: Krick-Logar-Kemper ( [15], [14]) and EisenbudHuneke-Vasconcelos ([8]). The results are shown in Table 1.…”

Section: Performance Evaluationmentioning

confidence: 99%

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An algorithm for the computation of the radical of an ideal

Laplagne

2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

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We propose a new algorithm for the computation of the radical of an ideal in a polynomial ring. In recent years many algorithms have been proposed. A common technique used is to reduce the problem to the zero dimensional case. In the algorithm we present here, we use this reduction, but we avoid the redundant components that appeared in other algorithms . As a result, our algorithm is in some cases more efficient.

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“…(c) Algorithms for the computation of radical ideals, as required in step (5), can be found in Becker and Weispfenning [2, Chapter 9], Derksen and Kemper [8, Section 1.5], and Kemper [14]. The latter two references provide algorithms for the case of positive characteristic which is relevant here.…”

Section: Computing the Normalizationmentioning

confidence: 99%