# Article

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Keywords:
polynomials in several variables over field; generalized polynomials in several variables over field; isomorphism of the ring of polynomials; automorphism of the ring of generalized polynomials; greatest common divisor of generalized polynomials
Summary:
On the ring \$R=F[x_1,\dots ,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,\dots ,x_n]]^+\$ and the ring \$T\:=F[[x_1,\dots ,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,\dots ,x_n]]\$ has a greatest common divisor.
References:
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