This paper investigates equivalence of square multivariate polynomial matrices with the determinant being some power of a univariate irreducible polynomial.The authors give a necessary and sufficient condition for thi...This paper investigates equivalence of square multivariate polynomial matrices with the determinant being some power of a univariate irreducible polynomial.The authors give a necessary and sufficient condition for this equivalence.And the authors present an algorithm to reduce a class of multivariate polynomial matrices to their Smith forms.展开更多
Gao,et al.(2015)gave a simple algorithm to compute Gr?bner bases named GVW.It 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 resolutio...Gao,et al.(2015)gave a simple algorithm to compute Gr?bner bases named GVW.It 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 algebra.In 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 top-reduction.In this paper,based on the study of relations between J-pairs,the authors propose the concept of factor.This concept allows the authors to filter the useless J-pairs in a rather convenient way.Moreover,by using this concept,the authors may easily determine which two pairs’J-pair need not to be computed.Besides,the Gr?bner basis which the authors obtained is relatively simpler than the one in GVW.展开更多
Multivariate(n-D)polynomial matrix factorization is one of important research contents in multidimensional(n-D)systems,circuits,and signal processing.In this paper,several results on n-D polynomial matrices factorizat...Multivariate(n-D)polynomial matrix factorization is one of important research contents in multidimensional(n-D)systems,circuits,and signal processing.In this paper,several results on n-D polynomial matrices factorization over arbitrary coefficient fields are proved.Based on these results,generalizations of some results on general matrix factorization are obtained for given n-D polynomial matrices whose maximal order minors or lower order minors satisfy certain conditions.The proposed results fit for arbitrary coefficient field and have a wide range of application.展开更多
GVW algorithm was given by Gao, Wang, and Volny in computing a Grobuer bases for ideal in a polynomial ring, which is much faster and more simple than F5. In this paper, the authors generalize GVW algorithm and presen...GVW algorithm was given by Gao, Wang, and Volny in computing a Grobuer bases for ideal in a polynomial ring, which is much faster and more simple than F5. In this paper, the authors generalize GVW algorithm and present an algorithm to compute a Grobner bases for ideal when the coefficient ring is a principal ideal domain. K展开更多
Gao, Volny and Wang （2010） gave a simple criterion for signature-based algorithms to compute GrSbner bases. It gives a unified frame work for computing GrSbner bases for both ideals and syzygies, the latter is very ...Gao, Volny and Wang （2010） gave a simple criterion for signature-based algorithms to compute GrSbner bases. It gives a unified frame work for computing GrSbner bases for both ideals and syzygies, the latter is very important in free resolutions in homological algebra. Sun and Wang （2011） later generalized the GVW criterion to a more general situation （to include the F5 Algorithm）. Signature-based algorithms have become increasingly popular for computing GrSbner bases. The current paper introduces a concept of factor pairs that can be used to detect more useless J-pairs than the generalized GVW criterion, thus improving signature-based algorithms.展开更多
基金supported by the Scientific Research Fund of Education Department of Hunan ProvinceChina under Grant Nos.20C0790 and 22A0334+1 种基金the National Natural Science Foundation of China under Grant Nos.11971161,12201204,and 12371507the Natural Science Foundation of Hunan Province under Grant No.2023JJ40275。
文摘This paper investigates equivalence of square multivariate polynomial matrices with the determinant being some power of a univariate irreducible polynomial.The authors give a necessary and sufficient condition for this equivalence.And the authors present an algorithm to reduce a class of multivariate polynomial matrices to their Smith forms.
基金supported by the National Natural Science Foundation of China under Grant No.11871207the General Project of Hunan Provincial Education Department under Grant No.17C0635the Natural Science Foundation of Hunan Provincial under Grant No.2017JJ3084。
文摘Gao,et al.(2015)gave a simple algorithm to compute Gr?bner bases named GVW.It 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 algebra.In 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 top-reduction.In this paper,based on the study of relations between J-pairs,the authors propose the concept of factor.This concept allows the authors to filter the useless J-pairs in a rather convenient way.Moreover,by using this concept,the authors may easily determine which two pairs’J-pair need not to be computed.Besides,the Gr?bner basis which the authors obtained is relatively simpler than the one in GVW.
基金supported by the National Natural Science Foundation of China under Grant Nos.11871207and 11971161the Natural Science Foundation of Hunan provincial under Grant No.2017JJ3084the Scientific Research Fund of Education Department of Hunan Province under Grant No.17C0635.
文摘Multivariate(n-D)polynomial matrix factorization is one of important research contents in multidimensional(n-D)systems,circuits,and signal processing.In this paper,several results on n-D polynomial matrices factorization over arbitrary coefficient fields are proved.Based on these results,generalizations of some results on general matrix factorization are obtained for given n-D polynomial matrices whose maximal order minors or lower order minors satisfy certain conditions.The proposed results fit for arbitrary coefficient field and have a wide range of application.
基金supported by the National Natural Science Foundation of China under Grant Nos.11071062,11271208Scientific Research Fund of Hunan Province Education Department under Grant Nos.10A033,12C0130
文摘GVW algorithm was given by Gao, Wang, and Volny in computing a Grobuer bases for ideal in a polynomial ring, which is much faster and more simple than F5. In this paper, the authors generalize GVW algorithm and present an algorithm to compute a Grobner bases for ideal when the coefficient ring is a principal ideal domain. K
基金supported by the National Natural Science Foundation of China under Grant Nos.11471108,11426101Hunan Provincial Natural Science Foundation of China under Grant Nos.14JJ6027,2015JJ2051Fundamental Research Funds for the Central Universities of Central South University under Grant No.2013zzts008
文摘Gao, Volny and Wang （2010） gave a simple criterion for signature-based algorithms to compute GrSbner bases. It gives a unified frame work for computing GrSbner bases for both ideals and syzygies, the latter is very important in free resolutions in homological algebra. Sun and Wang （2011） later generalized the GVW criterion to a more general situation （to include the F5 Algorithm）. Signature-based algorithms have become increasingly popular for computing GrSbner bases. The current paper introduces a concept of factor pairs that can be used to detect more useless J-pairs than the generalized GVW criterion, thus improving signature-based algorithms.
