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An Improvement of the rational representation for High-Dimensional Systems

Journal of Systems Science & Complexity 2021年第6期34卷 2410-2427页

作者： XIAO Fanghui LU Dong MA Xiaodong WANG Dingkang KLMM Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China Beijing Advanced Innovation Center for Big Data and Brain Computing Beihang UniversityBeijing 100191China School of Mathematical Sciences Beihang UniversityBeijing 100191China College of Science China Agricultural UniversityBeijing 100083China

Based on the rational univariate representation of zero-dimensional polynomial systems,Tan and Zhang proposed the rational representation theory for solving a high-dimensional polynomial system,which uses so-called rational representation sets to describe all the zeros of a high-dimensional polynomial *** paper is devoted to giving an improvement for the rational *** idea of this improvement comes from a minimal Dickson basis used for computing a comprehensive Grobner system of a parametric polynomial system to reduce the number of *** authors replace the normal Grobner basis G satisfying certain conditions in the original algorithm(Tan-Zhang’s algorithm)with a minimal Dickson basis G_(m) of a Grobner basis for the ideal,where G_(m) is smaller in size than *** on this,the authors give an improved ***,the proposed algorithm has been implemented on the computer algebra system *** data and its performance comparison with the original algorithm show that it generates fewer branches and the improvement is rewarding.

关键词： Comprehensive Grobner systems high-dimensional polynomial system rational representation rational univariate representation

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Computation of the rational representation for Solutions of High-dimensional Systems

引用

Communications in Mathematical Research 2010年第2期26卷 119-130页

作者： TAN CHANG ZHANG SHU-GONG Institute of Mathematics Jilin University Changchun 130012

This paper deals with the representation of the solutions of a polynomial system, and concentrates on the high-dimensional case. Based on the rational univari- ate representation of zero-dimensional polynomial systems, we give a new description called rational representation for the solutions of a high-dimensional polynomial sys- tem and propose an algorithm for computing it. By this way all the solutions of any high-dimensional polynomial system can be represented by a set of so-called rational- representation sets.

关键词： rational univariate representation high-dimensional ideal maximally independent set rational representation irreducible component

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