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Journal of Systems Science & Complexity 2022年第1期35卷 427-436页

作者： ZHENG Licui LI Dongmei LIU Jinwang Department of Mathematics and Computing Sciences Hunan University of Science and TechnologyXiangtan 411201China

Gao,et al.(2015)gave a simple algorithm to compute Gr?bner bases named *** can be used to compute Gr?bner bases for both ideals and syzygies at the same time,and the latter plays an important role in free resolutions in homological *** gvw algorithms the authors need to compute all the J-pairs firstly and then use gvw criterion(which refers the criterions used in gvw)to determine which one is useless or which one the authors should do *** this paper,based on the study of relations between J-pairs,the authors propose the concept of *** concept allows the authors to filter the useless J-pairs in a rather convenient ***,by using this concept,the authors may easily determine which two pairs’J-pair need not to be ***,the Gr?bner basis which the authors obtained is relatively simpler than the one in gvw.

关键词： Factor Grobner basic gvw

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Speeding Up the gvw Algorithm via a Substituting Method

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Journal of Systems Science & Complexity 2019年第1期32卷 205-233页

作者： LI Ting SUN Yao HUANG Zhenyu WANG Dingkang LIN Dongdai SKLOIS Institute of Information EngineeringChinese Academy of Sciences School of Cyber Security University of Chinese Academy of Sciences KLMM Academy of Mathematics and Systems ScienceChinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences

The gvw algorithm is an effcient signature-based algorithm for computing Gr?bner *** this paper, the authors consider the implementation of the gvw algorithm by using linear algebra,and speed up gvw via a substituting method. As it is well known that, most of the computing time of a Gr?bner basis is spent on reductions of polynomials. Thus, linear algebraic techniques, such as matrix operations, have been used extensively to speed up the implementations. Particularly, one-direction(also called signature-safe) reduction is used in signature-based algorithms, because polynomials(or rows in matrices) with larger signatures can only be reduced by polynomials(rows) with smaller signatures. The authors propose a new method to construct sparser matrices for signature-based algorithms via a substituting method. Speci?cally, instead of only storing the original polynomials in gvw, the authors also record many equivalent but sparser polynomials at the same time. In matrix construction, denser polynomials are substituted by sparser equivalent ones. As the matrices get sparser, they can be eliminated more effciently. Two speci?cal algorithms, Block-gvw and LMgvw, are presented, and their combination is the Sub-gvw algorithm. The correctness of the new proposed method is proved, and the experimental results demonstrate the effciency of this new method.

关键词： Gröbner basis gvw signature-based algorithm time-memory tradeoff

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