Determinants of Smith matrices on three coprime divisor chains and divisibility

Abstract

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 σ of {1, … , h} such that x σ(1)|…|x σ(h). We say that the set S consists of three coprime divisor chains if we can partition S as S = S 1 ∪ S 2 ∪ S 3, where S 1, S 2 and S 3 are divisor chains and each element of S i is coprime to each element of S j for all 1 ≤ i ≠ j ≤ 3. The matrix having the ath power (x i , x j ) a of the greatest common divisor of x i and x j as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (S a ). Similarly we can define the ath power LCM matrix [S a ]. In this article, we first obtain the formulae for determinants of power GCD matrices and power LCM matrices defined on three coprime divisor chains. Consequently, we establish several divisibility theorems among the determinants of these power matrices: if S consists of three coprime divisor chains and a|b, then under some natural conditions, we have det(S a )|det(S b ), det[S a ] |det[S b ] and det(S a )|det[S b ]. We also give an example which shows that such divisibility results are not true if those natural conditions are not satisfied.