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Hilbert Problem 15 and Ritt-Wu Method(Ⅱ)

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Journal of Systems Science & Complexity 2020年第6期33卷 2124-2138页

作者： LI Banghe wang dingkang KLMM Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu *** are relations of some special lines:1)For a given point,all the curves containing this point are *** any such curve,there are five *** of them are the tangent lines of the curve passing through the given *** other three are the lines connecting the given point with the cusp,the inflexion point and the intersection point of the tangent line at the cusp and the inflexion line.2)For a given point,the curves whose tangent line at the cusp passes through this point are *** any such curve,there are four *** of them are the tangent lines passing through this point and the other is the line connect the given point and the inflexion point.3)For a given point,the curves whose cusp,inflexion point and the given point are collinear are *** any such curve,there are five *** of them are tangent lines passing through the given *** other two are the lines connecting the given point with the cusp and the intersection point of the tangent line at the cusp and the inflexion line.

关键词： Cubic curves with cusp Hilbert problem 15 Ritt-Wu method

THE F5 ALGORITHM IN BUCHBERGER＇S STYLE

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Journal of Systems Science & Complexity 2011年第6期24卷 1218-1231页

作者： Yao SUN dingkang wang Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

The famous F5 algorithm for computing Grobner basis was presented by Faugere in 2002. The original version of F5 is given in programming codes, so it is a bit difficult to understand. In this paper, the F5 algorithm is simplified as F5B in a Buchberger＇s style such that it is easy to understand and implement. In order to describe F5B, we introduce F5-reduction, which keeps the signature of labeled polynomials unchanged after reduction. The equivalence between F5 and F5B is also shown. At last, some versions of the F5 algorithm are illustrated.

关键词： Buchberger's style F5 algorithm Grobner basis.

Solving the Perspective-Three-Point Problem Using Comprehensive Grobner Systems

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Journal of Systems Science & Complexity 2016年第5期29卷 1446-1471页

作者： ZHOU Jie wang dingkang Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China.

A complete solution classification of the perspective-three-point(P3P) problem is given by using the Gr?bner basis method. The structure of the solution space of the polynomial system deduced by the P3P problem can be obtained by computing a comprehensive Gr?bner system. Combining with properties of the generalized discriminant sequences, the authors give the explicit conditions to determine the number of distinct real positive solutions of the P3P problem. Several examples are provided to illustrate the effectiveness of the proposed conditions.

关键词： Comprehensive GrSbner system parametric polynomials perspective-three-point prob-lem real solutions.

On the Construction of Involutory MDS Matrices over F2^m

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Journal of Systems Science & Complexity 2020年第3期33卷 836-848页

作者： BAI Jian SUN Yao wang dingkang KLMM Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China SKLOIS Institute of Information EngineeringChinese Academy of SciencesBeijing 100093China

This paper studies the problem of constructing lightweight involutory maximal distance separable(MDS)*** authors find the exact lower bound of the XOR counts for 4×4 involutory MDS matrices over F2^***,some new structures of 4×4 involutory MDS matrices over F2^mare provided to construct involutory MDS matrices and the authors constructed the lightest4×4 involutory MDS matrices over F2^8so far by using these structures.

关键词： Diffusion layer involutory MDS matrix lightweight

Solving Multivariate Polynomial Matrix Diophantine Equations with Gr?bner Basis Method

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Journal of Systems Science & Complexity 2022年第1期35卷 413-426页

作者： XIAO Fanghui LU Dong wang dingkang College of Mathematics and Statistics Hunan Normal UniversityChangsha 410006China Beijing Advanced Innovation Center for Big Data and Brain Computing Beihang UniversityBeijing 100191China School of Mathematical Sciences Beihang UniversityBeijing 100191China. KLMM Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

Different from previous viewpoints,multivariate polynomial matrix Diophantine equations are studied from the perspective of modules in this paper,that is,regarding the columns of matrices as elements in modules.A necessary and sufficient condition of the existence for the solution of equations is *** powerful features and theoretical foundation of Gr?bner bases for modules,the problem for determining and computing the solution of matrix Diophantine equations can be ***,the authors make use of the extension on modules for the GVW algorithm that is a signature-based Gr?bner basis algorithm as a powerful tool for the computation of Gr?bner basis for module and the representation coefficients problem directly related to the particular solution of *** a consequence,a complete algorithm for solving multivariate polynomial matrix Diophantine equations by the Gr?bner basis method is presented and has been implemented on the computer algebra system Maple.

关键词： Gr?bner basis matrix Diophantine equation module multivariate polynomial

A New Algorithm for Computing the Extended Hensel Construction of Multivariate Polynomials

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Journal of Systems Science & Complexity 2018年第6期31卷 1633-1646页

作者： LU Dong SUN Yao wang dingkang KLMM Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China SKLOIS Institute of Information EngineeringChinese Academy of SciencesBeijing 100093China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China.

This paper presents a new algorithm for computing the extended Hensel construction(EHC) of multivariate polynomials in main variable x and sub-variables u1, u2, ···, um over a number field K. This algorithm first constructs a set by using the resultant of two initial coprime factors w.r.t. x, and then obtains the Hensel factors by comparing the coefficients of xi on both sides of an equation. Since the Hensel factors are polynomials of the main variable with coefficients in fraction field K(u1, u2, ···, um), the computation cost of handling rational functions can be high. Therefore,the authors use a method which multiplies resultant and removes the denominators of the rational functions. Unlike previously-developed algorithms that use interpolation functions or Grobner basis, the algorithm relies little on polynomial division, and avoids multiplying by different factors when removing the denominators of Hensel factors. All algorithms are implemented using Magma, a computational algebra system and experiments indicate that our algorithm is more efficient.

关键词： Extended Hensel construction multivariate polynomial resultant sylvester matrix

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

An efficient algorithm for factoring polynomials over algebraic extension field

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Science China Mathematics 2013年第6期56卷 1155-1168页

作者： SUN Yao wang dingkang SKLOIS Institute of Information Engineering Chinese Academy of Sciences KLMM Academy of Mathematics and Systems Science Chinese Academy of Sciences

An efficient algorithm is proposed for factoring polynomials over an algebraic extension field defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its CrSbner basis, no extra Grbbner basis computation is needed for factoring a polynomial over this extension field. Nothing more than linear algebraic technique is used to get a characteristic polynomial of a generic linear map. Then this polynomial is factorized over the ground field. From its factors, the factorization of the polynomial over the extension field is obtained. The algorithm has been implemented in Magma and computer experiments indicate that it is very efficient, particularly for complicated examples.

关键词： algorithm factorization algebraic extension field

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智能化的企业培训绩效评估策略分析

集成电路应用 2024年第1期41卷 194-195页

作者：王定康戴浩凯宁波送变电建设有限公司永耀科技分公司浙江315000

阐述智能化技术在绩效评估中的特点,分析智能化技术在绩效评估中提高评估的时效性和效率、提供更加个性化的评估和反馈措施、人力资源培养与智能化技术的融合策略。

关键词：智能化技术数据挖掘绩效评估