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

Skula, Ladislav. "Special isomorphisms of $F[x_1,\ldots ,x_n]$ preserving GCD and their use." Czechoslovak Mathematical Journal 59.3 (2009): 759-771. <http://eudml.org/doc/37956>.

@article{Skula2009,
abstract = {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.},
author = {Skula, Ladislav},
journal = {Czechoslovak Mathematical Journal},
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; polynomial over field; generalized polynomial over field; several variables; isomorphism of the ring of polynomials; automorphism; greatest common divisor of generalized polynomials},
language = {eng},
number = {3},
pages = {759-771},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Special isomorphisms of $F[x_1,\ldots ,x_n]$ preserving GCD and their use},
url = {http://eudml.org/doc/37956},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Skula, Ladislav
TI - Special isomorphisms of $F[x_1,\ldots ,x_n]$ preserving GCD and their use
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 3
SP - 759
EP - 771
AB - 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.
LA - eng
KW - 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; polynomial over field; generalized polynomial over field; several variables; isomorphism of the ring of polynomials; automorphism; greatest common divisor of generalized polynomials
UR - http://eudml.org/doc/37956
ER -