Power GCD Matrices for a UFD

Abstract

Let S = {x 1 ,…, x n } be a set of n distinct positive integers and e ≥ 1 an integer. Then we call the n × n matrix (S e ) = ((x i , x j ) e ) having the eth power of the greatest common divisor (x i , x j ) of x i and x j as its (i, j)-entry the power greatest common divisor (GCD) matrix on S. A well-known formula for det (S e ) when S = {1, …, n} was given by H.J.S. Smith in 1875/76. In this paper, we consider the power GCD matrices for a UFD along the direction of their structure, determinant and non-singularity, and consequently, we extend the classical result of Smith as well as the results of Beslin and Kassar on UFD given in 1989.