Power GCD Matrices for a UFD.

Let S = {x1,..., xn} be a set of n distinct positive integers and e ≥ 1 an integer. Then we call the n × n matrix (Se) = ((xi, xj)e) having the eth power of the greatest common divisor (xi, xj) of xi and xj as its (i, j)-entry the power greatest common divisor (GCD) matrix on S. A well-known formula for det(Se) 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. [ABSTRACT FROM AUTHOR]