Signature rewriting in gröbner basis computation

Eder, Christian; Roune, Bjarke Hammersholt

doi:10.1145/2465506.2465522

Cited by 10 publications

(26 citation statements)

References 8 publications

(16 reference statements)

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“…Still note that notations itself play an important role in the following, especially when comparing different variants of signature-based algorithms. We extend the notation introduced in [31].…”

Section: Notations and Terminologymentioning

confidence: 99%

“…The algorithms presented there, called TRB and GBGC are included in Algorithm 2, called RB. Note that in [31] there is already an algorithm called RB, here we generalized it further.…”

Section: Rewrite Basesmentioning

confidence: 99%

“…Analogously to Section 3.2 in [31] we introduce next the important notion of a rewrite basis. Note that the combination of the syzygy and the singular criterion lead to a much easier notation.…”

Section: Combining Elimination Criteriamentioning

confidence: 99%

“…Moreover, first papers trying to classify all the different variants of signature-based Gröbner basis algorithms came up [30,31,58,68,69,76].…”

Section: Introductionmentioning

confidence: 99%

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A survey on signature-based algorithms for computing Gröbner bases

Eder

Faugère

2017

Journal of Symbolic Computation

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This paper is a survey on the area of signature-based Gröbner basis algorithms that was initiated by Faugère's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research. At the moment the area of signature-based Gröbner basis algorithms is confusing and vast. More and more papers are published proving statements already proven before, and even more publications can be found on "new" variants that boil down to be a known one just with a different notation. In this paper we try to give a rigorous survey on signature-based Gröbner basis theory, including all variants known up to now. We lay an emphasis on understanding and we show how the variants presented over the last years are mostly differ in small parts only. Moreover, we give the reader a vocabulary book at hand which helps to understand how notations, varying for different authors, coincide. Since this is a survey, we do not give proofs if they are long, complex, or do not help in understanding the topic. We always explain the idea behind the proofs and refer to the related publication which includes a complete proof. There the reader is then, with our descriptions and explanations, able to understand the

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Section: Notations and Terminologymentioning

confidence: 99%

“…The algorithms presented there, called TRB and GBGC are included in Algorithm 2, called RB. Note that in [31] there is already an algorithm called RB, here we generalized it further.…”

Section: Rewrite Basesmentioning

confidence: 99%

Section: Combining Elimination Criteriamentioning

confidence: 99%

“…Moreover, first papers trying to classify all the different variants of signature-based Gröbner basis algorithms came up [30,31,58,68,69,76].…”

Section: Introductionmentioning

confidence: 99%

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A survey on signature-based algorithms for computing Gröbner bases

Eder

Faugère

2017

Journal of Symbolic Computation

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“…We refer to the corresponding papers for more details. [18] and [19] give an overview on how the 4 variants are related to each other. Note that combining F5 with ≺ s does not introduce any theoretical problems for correctness of the algorithm.…”

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confidence: 99%

An analysis of inhomogeneous signature-based Gröbner basis computations

Eder

2013

Journal of Symbolic Computation

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In this paper we give an insight into the behaviour of signature-based Gröbner basis algorithms, like F5, G2V or SB, for inhomogeneous input. On the one hand, it seems that the restriction to sig-safe reductions puts a penalty on the performance. The lost connection between polynomial degree and signature degree can disallow lots of reductions and can lead to an overhead in the computations. On the other hand, the way critical pairs are sorted and corresponding s-polynomials are handled in signaturebased algorithms is a very efficient one, strongly connected to sorting w.r.t. the well-known sugar degree of polynomials.

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A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains

Francis

Verron

2019

Math.Comput.Sci.

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Signature-based algorithms are the latest and most efficient approach as of today to compute Gröbner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention of several authors.In this paper, we present a signature-based version of Möller's classical variant of Buchberger's algorithm for computing strong Gröbner bases over Principal Ideal Domains (or PIDs). It ensures that the signatures do not decrease during the algorithm, which makes it possible to apply classical signature criteria for further optimization. In particular, with the F5 criterion, the signature version of Möller's algorithm computes a Gröbner basis without reductions to zero for a polynomial system given by a regular sequence. We also show how Buchberger's chain criterion can be implemented so as to be compatible with the signatures.We prove correctness and termination of the algorithm. Furthermore, we have written a toy implementation in Magma, allowing us to quantitatively compare the efficiency of the various criteria for eliminating S-pairs.

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Signature rewriting in gröbner basis computation

Cited by 10 publications

References 8 publications

A survey on signature-based algorithms for computing Gröbner bases

A survey on signature-based algorithms for computing Gröbner bases

An analysis of inhomogeneous signature-based Gröbner basis computations

A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains

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