GCD封闭集上幂矩阵行列式的整除性  

Divisibility among determinants of power matrices on GCD-closed sets

作　　者：朱光艳 强诗瑗 林宗兵 ZHU Guang-Yan;QIANG Shi-Yuan;LIN Zong-Bing(School of Teacher Education,Hubei Minzu University,Enshi 445000,China;School of Mathematics,Sichuan University,Chengdu 610064,China;School of Mathematics and Computer Science,Panzhihua University,Panzhihua 617000,China)

机构地区：[1]湖北民族大学教师教育学院,恩施445000 [2]四川大学数学学院,成都610064 [3]攀枝花学院数学与计算机学院,攀枝花617000

出　　处：《四川大学学报（自然科学版）》2023年第5期79-84,共6页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(11771304);攀枝花学院博士基金(bkqj2019050)。

摘　　要：设S={x_(1),…,xn}为n个不同正整数构成的集合,若对任意不超过n的正整数i,j,均有gcd(x_(i),x_(j))∈S,则称S是GCD封闭集.对于元素x,y∈S(y<x),若由y|z|x和z∈S可推出z∈{y,x},则称y是x的一个最大型因子.令GS(x)表示x在S中所有最大型因子构成的集合.设a和b是正整数,f是算术函数.以(f^(a)(S))(对应地(f^(a)[S]))表示一个n阶方阵,其第i行第j列元素为f^(a)(gcd(x_(j),x_(j)))(对应地f^(a)(lcm(x_(j),x_(j)))).令|T|表示有限集T的基数.在本文中,当a|b,S为GCD封闭集且maxx∈S{|GS(x)|}≤2时,我们建立了几个关于幂矩阵(f^(a)(S))与(f^(b)(S)),(f^(a)(S))与(f^(b)[S]),(f^(a)[S])与(f^(b)[S])的行列式之间的整除性结果.Let S={x_(1),…,x_(n)}be a set of n distinct positive integers x_(1),…,x_(n),which is called GCD closed if gcd(x_(i),x_(j))∈S for all integers i,j with 1≤i,j≤n.For x,y∈S,if the conditions yx,y|z|x and z∈S imply that z∈{y,x},then y is called a greatest-type divisor of x in S.Let G_S(x)denote the set of all greatest-type divisors of x in S.Let a and b be positive integers.Denote by(f^(a)(S))and(f^a[S])the n×n matrix with f^(a)evaluated at the greatest common divisor and the least common multiple of x_(i)and x_(j)as its(i,j)-entry,respectively.Let|T|denote the cardinal number of a finite set T.In this paper,we establish some results on the divisibility among determinants of power matrices(f^a(S))and(f^b(S)),(f^a(S))and(f^(b)[S]),(f^(a)[S])and(f^(b)[S]),where a|b,S is GCD closed and max_(x∈S){|G_S(x)|}≤2.

关 键 词：整除 算术函数 幂矩阵 GCD封闭集 

分 类 号：O156.1[理学—数学]