On implementing the symbolic preprocessing function over Boolean polynomial rings in Gröbner basis algorithms using linear algebra

Sun, Yichuang; Huang, Zhenyu; Lin, Dongdai; Wang, Dingkang

doi:10.1007/s11424-015-4085-1

Cited by 5 publications

(2 citation statements)

References 5 publications

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“…For these cases, instead of traversing objects in the large set, we can find out whether the desired object lies in the large set by using a hash table. Similar method is used in [23].…”

Section: Other Detailsmentioning

confidence: 99%

“…structure to the algorithm presented in [1]. Mo-GVW uses a frame like XL and avoids generating J-pairs, and mo-GVW also uses an improved method of constructing matrices [23], so mo-GVW outperforms M-GVW. Second, the elimination of matrices in present implementation of mo-GVW is mainly done by linear algebraic routines for dense matrices.…”

Section: On Implementing the Mo-gvw Algorithmmentioning

confidence: 99%

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A Monomial-Oriented GVW for Computing Gröbner Bases

Sun,

Wang,

Huang

et al. 2014

Preprint

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The GVW algorithm, presented by Gao et al., is a signature-based algorithm for computing Gröbner bases. In this paper, a variant of GVW is presented. This new algorithm is called a monomial-oriented GVW algorithm or mo-GVW algorithm for short. The mo-GVW algorithm presents a new frame of GVW and regards labeled monomials instead of labeled polynomials as basic elements of the algorithm. Being different from the original GVW algorithm, for each labeled monomial, the mo-GVW makes efforts to find the smallest signature that can generate this monomial. The mo-GVW algorithm also avoids generating J-pairs, and uses efficient methods of searching reducers and checking criteria. Thus, the mo-GVW algorithm has a better performance during practical implementations.

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“…For these cases, instead of traversing objects in the large set, we can find out whether the desired object lies in the large set by using a hash table. Similar method is used in [23].…”

Section: Other Detailsmentioning

confidence: 99%

Section: On Implementing the Mo-gvw Algorithmmentioning

confidence: 99%

A Monomial-Oriented GVW for Computing Gröbner Bases

Sun,

Wang,

Huang

et al. 2014

Preprint

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Speeding Up the GVW Algorithm via a Substituting Method

Sun

Huang

et al. 2019

J Syst Sci Complex

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An improvement over the GVW algorithm for inhomogeneous polynomial systems

Sun

Huang

Wang

et al. 2016

Finite Fields and Their Applications

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The GVW algorithm is a signature-based algorithm for computing Gröbner bases. If the input system is not homogeneous, some J-pairs with higher signatures but lower degrees are rejected by GVW's Syzygy Criterion, instead, GVW have to compute some J-pairs with lower signatures but higher degrees. Consequently, degrees of polynomials appearing during the computations may unnecessarily grow up higher and the computation become more expensive. In this paper, a variant of the GVW algorithm, called M-GVW, is proposed and mutant pairs are introduced to overcome inconveniences brought by inhomogeneous input polynomials. Some techniques from linear algebra are used to improve the efficiency. Both GVW and M-GVW have been implemented in C++ and tested by many examples from boolean polynomial rings. The timings show M-GVW usually performs much better than the original GVW algorithm when mutant pairs are found. Besides, M-GVW is also compared with intrinsic Gröbner bases functions on Maple, Singular and Magma. Due to the efficient routines from the M4RI library, the experimental results show that M-GVW is very efficient.

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On implementing the symbolic preprocessing function over Boolean polynomial rings in Gröbner basis algorithms using linear algebra

Cited by 5 publications

References 5 publications

A Monomial-Oriented GVW for Computing Gröbner Bases

A Monomial-Oriented GVW for Computing Gröbner Bases

Speeding Up the GVW Algorithm via a Substituting Method

An improvement over the GVW algorithm for inhomogeneous polynomial systems

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