Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains II

作者：Tan Qianrong^*; Lin Zongbing; Liu Liu

来源：Linear and Multilinear Algebra, 2011, 59(9): 969-983.

DOI：10.1080/03081087.2010.509721

• divisibility
• coprime divisor chains
• greatest-type divisor
• power GCD matrix
• power LCM matrix

Let a, b and h be positive integers and S = {x(1),...,x(h)} be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation sigma of {1,...,h} such that x(sigma(1))vertical bar...vertical bar x(sigma(h)). We say that the set S consists of two coprime divisor chains if we can partition S as S = S(1) boolean OR S(2), where S(1) and S(2) are divisor chains and each element of S(1) is coprime to each element of S(2). The matrix having the ath power (x(i), x(j))(a) of the greatest common divisor (GCD) of x(i) and x(j) as its (i, j)-entry is called the ath power GCD matrix defined on S, denoted by (S(a)). Similarly, we can define the ath power least common multiple (LCM) matrix [S(a)]. In the first paper of the series, Tan [Q. Tan, Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains, Linear Multilinear Algebra 58 (2010), pp. 659-671] showed that if S consists of two coprime divisor chains and 1 is an element of S and a vertical bar b, then (S(a)) j(S(b)), [S(a)]vertical bar[S(b)] and (S(a))vertical bar[S(b)] hold in the ring M(h)(Z) of h x h matrices over integers. But such factorizations need not hold if a x b. In this second paper of the series, we assume that S consists of two coprime divisor chains and 1 is not an element of 2 S. We show the following results: (i) If a x b, then (S(a)) x (S(b)), [S(a)] x [S(b)] and (S(a)) x [S(b)]. (ii) If a vertical bar b, then det(S(a)) vertical bar det(S(b)), det[S(a)] vertical bar det[S(b)] and det(S(a)) vertical bar det[S(b)]. (iii) If a vertical bar b, then (S(a)) vertical bar (S(b)), [S(a)] vertical bar [S(b)] and (S(a)) vertical bar [S(b)] hold in the ring M(h)(Z) if and only if both x(a)y(b)-1/x(a)y(a)-1 and x(b)y(a)-1/x(a)y(a)-1 are integers, where S = S(1) boolean OR S(2) with S(1) and S(2) divisor chains and x = min(S(1)) and y = min(S(2)). Our results extend Hong's results and complement Tan's results.

• 出版日期2011
• 单位四川大学; 攀枝花学院