Special isomorphisms of $F[x_1,\ldots ,x_n]$ preserving GCD and their use

Title:

Special isomorphisms of $F[x_1,\ldots ,x_n]$ preserving GCD and their use

Creator:

Skula, Ladislav

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:a949026f-f823-4b42-b831-9628c6d6f867
uuid:a949026f-f823-4b42-b831-9628c6d6f867

Subject:

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, and greatest common divisor of generalized polynomials

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

759-771

Source:

Czechoslovak Mathematical Journal | 2009 Volume:59 | Number:3

Harvested from:

CDK

Metadata only:

false