多项式组的特征分解  

Characteristic decomposition of polynomial sets

作　　者：王东明[1,2,3] 牟晨琪 董日娜 Dongming Wang;Chenqi Mou;Rina Dong

机构地区：[1]北京航空航天大学数学科学学院,数学、信息与行为教育部重点实验室,北京100191 [2]北京航空航天大学大数据科学与脑机智能高精尖创新中心,北京100191 [3]Centre National de la Recherche Scientifique,Paris 75794,France

出　　处：《中国科学：数学》2021年第1期67-86,共20页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11771034和11971050)资助项目。

摘　　要：任一多项式理想的特征对是指由该理想的约化字典序Grobner基G和含于其中的极小三角列C构成的有序对(G,C).当C为正则列或正规列时,分别称特征对(G,C)为正则的或正规的.当G生成的理想与C的饱和理想相同时,称特征对(G,C)为强的.一组多项式的(强)正则或(强)正规特征分解是指将该多项式组分解为有限多个(强)正则或(强)正规特征对,使其满足特定的零点与理想关系.本文简要回顾各种三角分解及相应零点与理想分解的理论和方法,然后重点介绍(强)正则与(强)正规特征对和特征分解的性质,说明三角列、Ritt特征列和字典序Grobner基之间的内在关联,建立特征对的正则化定理以及正则、正规特征对的强化方法,进而给出两种基于字典序Grobner基计算、按伪整除关系分裂和构建、商除可除理想等策略的(强)正规与(强)正则特征分解算法.这两种算法计算所得的强正规与强正则特征对和特征分解都具有良好的性质,且能为输入多元多项式组的零点提供两种不同的表示.本文还给出示例和部分实验结果,用以说明特征分解方法及其实用性和有效性.An ordered pair(G,C)is called the characteristic pair of a polynomial ideal if G is the reduced lexicographic Grobner basis of the ideal and C is the minimal triangular set contained in G.A characteristic pair(G,C)is said to be regular or normal if C is regular or normal;it is said to be strong if the ideal generated by G and the saturated ideal of C are the same.A strong regular or normal characteristic decomposition of a polynomial set means a decomposition of the polynomial set into nitely many(strong)regular or normal characteristic pairs with an associated zero or ideal decomposition.This paper presents two algorithms for characteristic decomposition of any given polynomial set,one producing strong normal characteristic pairs with ideal splitting according to pseudo-divisibility relations between polynomials in reduced lexicographic Grobner bases and the other producing strong regular characteristic pairs by constructing nontrivial ideals that can be divided out of given ideals by using ideal quotient.Novel theorems are established for regularizing characteristic pairs and strengthening regular and normal characteristic pairs by using ideal saturation.Strong regular or normal characteristic pairs produced by the algorithms possess nice properties and furnish two representations,one in terms of lexicographic Grobner bases and the other in terms of triangular sets,for the zeros of the input set of multivariate polynomials.The paper includes a brief review on triangular decompositions of various kinds and discussions about inherent connections among triangular sets,Ritt characteristic sets,and reduced lexicographic Grobner bases.Examples and experimental results are also provided to illustrate the method of characteristic decomposition and its applicability and performance.

关 键 词：特征对 特征分解 三角分解 GROBNER基 多项式组 

分 类 号：O174.14[理学—数学]