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A polynomial-time algorithm to compute generalized Hermite normal forms of matrices over Z[x]
Jing, Rui-Juan; Yuan, Chun-Ming; Gao, Xiao-Shan
2019-01-10
Source PublicationTHEORETICAL COMPUTER SCIENCE
ISSN0304-3975
Volume755Pages:89-109
AbstractIn this paper, we give the first polynomial time algorithm to compute the generalized Hermite normal form for a matrix F over Z[x], or equivalently, the reduced Grobner basis of the Z[x]-module generated by the column vectors of F. The algorithm has polynomial bit size computational complexities and is also shown to be practically more efficient than existing algorithms. The algorithm is based on three key ingredients. First, an F4 style algorithm to compute the Grobner basis is adopted, where a novel prolongation is designed such that the sizes of coefficient matrices under consideration are nicely controlled. Second, the complexity bound of the algorithm is achieved by a nice estimation for the degree and height bounds of the polynomials in the generalized Hermite normal form. Third, fast algorithms to compute Hermite normal forms of matrices over Z are used as the computational tool. (C) 2018 Elsevier B.V. All rights reserved.
KeywordGeneralized Hermite normal form Grobner basis Polynomial-time algorithm Z[x] module
DOI10.1016/j.tcs.2018.07.003
Language英语
Funding ProjectNSFC[11688101]
WOS Research AreaComputer Science
WOS SubjectComputer Science, Theory & Methods
WOS IDWOS:000456359300008
PublisherELSEVIER SCIENCE BV
Citation statistics
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/32138
Collection系统科学研究所
AffiliationChinese Acad Sci, Acad Math & Syst Sci, UCAS, KLMM, Beijing 100190, Peoples R China
Recommended Citation
GB/T 7714
Jing, Rui-Juan,Yuan, Chun-Ming,Gao, Xiao-Shan. A polynomial-time algorithm to compute generalized Hermite normal forms of matrices over Z[x][J]. THEORETICAL COMPUTER SCIENCE,2019,755:89-109.
APA Jing, Rui-Juan,Yuan, Chun-Ming,&Gao, Xiao-Shan.(2019).A polynomial-time algorithm to compute generalized Hermite normal forms of matrices over Z[x].THEORETICAL COMPUTER SCIENCE,755,89-109.
MLA Jing, Rui-Juan,et al."A polynomial-time algorithm to compute generalized Hermite normal forms of matrices over Z[x]".THEORETICAL COMPUTER SCIENCE 755(2019):89-109.
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