有理数域上一类多项式可约性的一个判别法  

A Criterion for the Reducibility of a Class of Integer Polynomials over the Field of Rational Numbers

作　　者：赵世忠[1] 符红光[2] 秦小林[3] 刘静[1] 刘云浩 ZHAO Shizhong;FU Hongguang;QIN XiaoLin;LIU Jing;LIU YunHao(Shanghai Key Laboratory of Trustworthy Computing,Software Engineering Institute,East China Normal University,Shanghai 200062;School of Computer Science and Engineering,University of Electronic Science and Technology of China,Chengdu 611731;Chengdu Institute of Computer Applications,Chinese Academy of Sciences,Chengdu 610041)

机构地区：[1]华东师范大学软件工程学院,上海市高可信计算重点实验室,上海200062 [2]电子科技大学计算机科学与工程学院,成都611731 [3]中国科学院成都计算机应用研究所,成都610041

出　　处：《系统科学与数学》2021年第12期3351-3362,共12页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金(12171159,61972150,61876034);国家“863”计划课题(2018YFB1005100,2018YFB1005104);国家重点研发计划(2019YFA0706404);中国科学院“西部青年学者”项目(201899);四川省委组织部人才专项资助课题。

摘　　要：任给一个m次的整系数多项式∑_(i=0)^(m)aix^(i),其中首项系数am=1,以及对应的下列不动点迭代算法{u1=u^(~)1,u2=u^(~)2,…u_(m-1)=u^(~)_(m-1),u_(n)=-(a_(m-1)+a_(m-2)/u_(n-1)+a_(m-3)/u_(n-1)u_(n-2)+…+a0/u_(n-1)u_(n-2)…u_(n-(m-1)))(n≥m).1)不难看出,若迭代具有一个有理数极限值,则该值为多项式的一个零点,从而多项式在有理数域上可约.2)该迭代具有"勿需选择初始点"的特征:若多项式有m个绝对值互不相同的有理数零点,那么任意取m-1个非零有理初始点(u)i(1≤i≤m-1),迭代均趋近于其中一个零点,因此,多项式可约.3)假设{ζ_(i)||ζ_(1)|≥|ζ_(2)|≥...≥|ζ_(m)|,ζ_(i)∈C,1≤i≤m}是上述多项式互不相同的零点,则存在m个复数{β_(i)|β_(i)∈C,1≤i≤m},使得u_(n)可以表示成u_(n)=β_(1)ζ_(1)^(n+1)+β_(2)ζ_(2)^(n+2)+…+β_(m)ζ_(m)^(n+1)/β_(1)ζ_(1)^(n)+β_(2)ζ_(2)^(n)+…+β_(m)ζ_(m)^(n)在β向量的m个元素中,设βl是首个非0元素,βk为其后首个非0元素,即{β_(i)|β_(i)∈C,1≤i≤m}={(0,0,…,0,)全为0βl(≠0),(0,0,…,0,)全为0β_(k)(≠0),...,β_(m)}.这时,若|ζ_(l)|>|ζk|,则迭代收敛于ζ_(l).因此,若ζ_(l)∈Q,则多项式可约.For the integer polynomials ∑_(i=0)^(m)aix^(i)of degree m,where a;=1,this paper presents a fixed-point iterative algorithm:{u1=u^(~)1,u2=u^(~)2,…u_(m-1)=u^(~)_(m-1),u_(n)=-(a_(m-1)+a_(m-2)/u_(n-1)+a_(m-3)/u_(n-1)u_(n-2)+…+a0/u_(n-1)u_(n-2)…u_(n-(m-1)))(n≥m).1) Obviously,if the iteration has a rational limit value,then the value is a zero of the polynomial,so that the polynomial is reducible over Q.2) This iteration does not need to choose the initial values:If the polynomial has m rational zeros with different absolute values,then for any m-1 non-zero rational initial values u;(1≤i≤m-1),the iteration approaches to one of the zeros.Therefore,the polynomial is reducible.3)Assuming that {ζ_(i)||ζ_(1)|≥|ζ_(2)|≥...≥|ζ_(m)|,ζ_(i)∈C,1≤i≤m} are the distinct zeros of the above polynomial,there exist m complex numbers {β_(i)|β_(i)∈C,1≤i≤m} such that un can be expressed in the following form u_(n)=β_(1)ζ_(1)^(n+1)+β_(2)ζ_(2)^(n+2)+…+β_(m)ζ_(m)^(n+1)/β_(1)ζ_(1)^(n)+β_(2)ζ_(2)^(n)+…+β_(m)ζ_(m)^(n).Among the m elements of the vector β,let β;be the first non-zero element and β;the first non-zero element after it,that is,{β_(i)|β_(i)∈c,1≤i≤m}=■β_(l)(≠0),…,β_(m)}.In this case,if |ζ_(l)|>|ζ_(l)|,then the iteration converges to ζ_(l).Therefore,if ζ_(l)∈2,then the polynomial is reducible.

关 键 词：可约 不可约 多项式 迭代 零点 根 初始值 

分 类 号：O174.14[理学—数学]