Efficient Computations of Irredundant Triangular Decompositions with the RegularChains Library

Chen, Changbo; Lemaire, François; Maza, Marc Moreno; Pan, Wei; Xie, Y.

doi:10.1007/978-3-540-72586-2_38

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…Unfortunately, such computations can be expensive (see [3]) whereas one would like to obtain an inclusion test which could be used intensively in order to remove redundant components when computing the triangular decompositions of Kalkbrener's algorithm or those arising in differential algebra. Note that for other kinds of triangular decompositions, such as those of [17,20], this question has been solved in [6].…”

Section: Introductionmentioning

confidence: 98%

When does ( T ) equal sat( T )?

Lemaire

Maza

Pan

et al. 2008

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

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Given a regular chain T , we aim at finding an efficient way for computing a system of generators of sat(T ), the saturated ideal of T . A natural idea is to test whether the equality T = sat(T ) holds, that is, whether T generates its saturated ideal. By generalizing the notion of primitivity from univariate polynomials to regular chains, we establish a necessary and sufficient condition, together with a Gröbner basis free algorithm, for testing this equality. Our experimental results illustrate the efficiency of this approach in practice.

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Section: Introductionmentioning

confidence: 98%

When does ( T ) equal sat( T )?

Lemaire

Maza

Pan

et al. 2008

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

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“…The technique to compute triangular sets has been refined (cf. [DMSWX05], [CLMPX07], [LMX06]), modularized and parallelized (cf. [MX07], [LM07]).…”

Section: Introductionmentioning

confidence: 99%

Solving via Modular Methods

Afzal

Janjua

Pfister

et al. 2014

Bridging Algebra, Geometry, and Topology

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Abstract. In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using Möller's algorithm (cf.[Mö93]) of the corresponding ideal in the polynomial ring over the rationals using modular methods, and then apply a solver for univariate polynomials.

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“…for computing the maximal elements of the poset (V, ⊆). However, this problem has received little attention in the literature [6] since the questions attached to algebraic sets (like decomposing polynomial systems) are of much more complex nature.…”

Section: Introductionmentioning

confidence: 99%

Parallel computation of the minimal elements of a poset

Leiserson

Maza

et al. 2010

Proceedings of the 4th International Workshop on Parallel and Symbolic Computation

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Computing the minimal elements of a partially ordered finite set (poset) is a fundamental problem in combinatorics with numerous applications such as polynomial expression optimization, transversal hypergraph generation and redundant component removal, to name a few. We propose a divide-and-conquer algorithm which is not only cache-oblivious but also can be parallelized free of determinacy races. We have implemented it in Cilk++ targeting multicores. For our test problems of sufficiently large input size our code demonstrates a linear speedup on 32 cores.

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Efficient Computations of Irredundant Triangular Decompositions with the RegularChains Library

Cited by 3 publications

References 9 publications

When does ( T ) equal sat( T )?

When does ( T ) equal sat( T )?

Solving via Modular Methods

Parallel computation of the minimal elements of a poset

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