An algorithm for the computation of the radical of an ideal in the ring of polynomials

Krick, Teresa; Logar, Alessandro

doi:10.1007/3-540-54522-0_108

Cited by 57 publications

(38 citation statements)

References 10 publications

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“…If I is zero dimensional, there are well-known methods for computing √ I. For an ideal of positive dimension, various algorithms computing √ I have been proposed (Wu, 1984;Gianni et al, 1988;Krick and Logar, 1991;Eisenbud et al, 1992;Wang, 1993;Armendáriz and Pablo, 1995;Caboara et al, 1997;Wang, 1998). All of them successfully compute the radical of an ideal when the characteristic is sufficiently large (or 0).…”

Section: Introductionmentioning

confidence: 98%

Computing the Radical of an Ideal in Positive Characteristic

Matsumoto

2001

Journal of Symbolic Computation

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We propose a method for computing the radical of an arbitrary ideal in the polynomial ring in n variables over a perfect field of characteristic p > 0. In our method Buchberger's algorithm is performed once in n variables and a Gröbner basis conversion algorithm is performed at most n log p d times in 2n variables, where d is the maximum of total degrees of generators of the ideal and 3. Next we explain how to compute radicals over a finitely generated coefficient field over a field K, when we have a radical computation method over the field K. Thus we can compute radicals over any finitely generated field over a perfect field.

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Section: Introductionmentioning

confidence: 98%

Computing the Radical of an Ideal in Positive Characteristic

Matsumoto

2001

Journal of Symbolic Computation

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“…A systematic study of consequences that can be derived this way i s contained in [12]. Here we generalize these ideas to submodules of a nitely generated free S-module F, extending the results of [17] i n to a more computational direction.…”

Section: Reduction To Dimension Zeromentioning

confidence: 95%

Minimal Primary Decomposition and Factorized Gröbner Bases

Gräbe

1997

Applicable Algebra in Engineering, Communication and Computing

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This paper continues our study of applications of factorized Grobner basis computations in [8] and [9].We describe a way t o i n terweave factorized Grobner bases and the ideas in [5] that leads to a signi cant speed up in the computation of isolated primes for well splitting examples.Based on that observation we generalize the algorithm presented in [22] t o t h e computation of primary decompositions for modules. It rests on an ideal separation argument.We also discuss the practically important question how to extract a minimal primary decomposition, neither addressed in [5] nor in [17]. For that purpose we outline a method to detect necessary embedded primes in the output collection of our algorithm, similar to [22, cor. 2.22].The algorithms are partly implemented in version 2.2.1 of our REDUCE package CALI [7].

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“…The increasing availability of symbolic algebra systems on computers and of efficient methods for 1)-3) has led to a renewed interest in the question of computing primary decompositions, as one sees from the work of Lazard (1982 and1985), Gianni et al (1988) (see also the references there), Bayer et al (1992), and Krick and Logar (1991). However these authors make use of the same basic strategy as Hermann, using PROJECTION to reduce to the one-polynomial case as before.…”

Section: ) Find the Polynomial Solutions To Linear Equations With Pomentioning

confidence: 99%