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Generic Regular Decompositions for parametric polynomial systems

Journal of systems Science & Complexity 2015年第5期28卷 1194-1211页

作者： CHEN Zhenghong TANG Xiaoxian XIA Bican School of Mathematical Sciences Peking University

This paper presents a generalization of the authors' earlier work. In this paper, the two concepts, generic regular decomposition(GRD) and regular-decomposition-unstable(RDU) variety introduced in the authors' previous work for generic zero-dimensional systems, are extended to the case where the parametric systems are not necessarily zero-dimensional. An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in the authors' previous work. Then the solutions of any parametric system can be represented by the solutions of finitely many regular systems and the decomposition is stable at any parameter value in the complement of the associated RDU variety of the parameter space. The related definitions and the results presented in the authors' previous work are also generalized and a further discussion on RDU varieties is given from an experimental point of view. The new algorithm has been implemented on the basis of DISCOVERER with Maple 16 and experimented with a number of benchmarks from the literature.

关键词： Generic regular decomposition parametric polynomial system regular-decompositionunstable variety

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Generic Regular Decompositions for parametric polynomial systems

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Journal of systems Engineering and Electronics 2015年第5期26卷 1194-1211页

作者： CHEN Zhenghong TANG Xiaoxian XIA Bican School of Mathematical Sciences Peking University

This paper presents a generalization of the authors＇ earlier work. In this paper, the two concepts, generic regular decomposition （GRD） and regular-decomposition-unstable （RDU） variety introduced in the authors＇ previous work for generic zero-dimensional systems, are extended to the case where the parametric systems are not necessarily zero-dimensional. An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in the authors＇ previous work. Then the solutions of any parametric system can be represented by the solutions of finitely many regular systems and the decomposition is stable at any parameter value in the complement of the associated RDU variety of the parameter space. The related definitions and the results presented in the authors＇ previous work are also generalized and a further discussion on RDU varieties is given from an experimental point of view. The new algorithm has been implemented on the basis of DISCOVERER with Maple 16 and experimented with a number of benchmarks from the literature.

关键词： Generic regular decomposition parametric polynomial system regular-decomposition-unstable variety

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A Survey on Algorithms for Computing Comprehensive Gröbner systems and Comprehensive Gröbner Bases

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Journal of systems Science & Complexity 2019年第1期32卷 234-255页

作者： LU Dong SUN Yao WANG Dingkang KLMM Academy of Mathematics and Systems ScienceChinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences SKLOIS Institute of Information EngineeringChinese Academy of Sciences

Weispfenning in 1992 introduced the concepts of comprehensive Gr?bner system/basis of a parametric polynomial system, and he also presented an algorithm to compute them. Since then,this research ?eld has attracted much attention over the past several decades, and many effcient algorithms have been proposed. Moreover, these algorithms have been applied to many different ?elds,such as parametric polynomial equations solving, geometric theorem proving and discovering, quanti?er elimination, and so on. This survey brings together the works published between 1992 and 2018, and we hope that this survey is valuable for this research area.

关键词： Comprehensive Gröbner basis comprehensive Gröbner system discovering geometric theorems mechanically parametric polynomial system quantifier elimination

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