Special isomorphisms of F[x 1,..., x n ] preserving GCD and their use

Abstract

On the ring R = F[x 1,..., x n ] of polynomials in n variables over a field F special isomorphisms A’s of R into R are defined which preserve the greatest common divisor of two polynomials. The ring R is extended to the ring S: = F[[x 1,..., x n ]]+ and the ring T: = F[[x 1,..., x n ]] of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms A’s are extended to automorphisms B’s of the ring S. Using the property that the isomorphisms A’s preserve GCD it is shown that any pair of generalized polynomials from S has the greatest common divisor and the automorphisms B’s preserve GCD. On the basis of this Theorem it is proved that any pair of generalized polynomials from the ring T = F[[x 1,..., x n ]] has a greatest common divisor.