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

Ladislav Skula

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 3, page 759-771
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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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 -

References

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  1. Karásek, J., Šlapal, J., Polynomials and Generalized Polynomials for the Theory of Control. Special Monograph, Academic Publishing House CERM Brno (2007), Czech. (2007) 
  2. Nicholson, W. K., Introduction to Abstract Algebra, PWS-KENT Publishing Company Boston (1993). (1993) Zbl0781.12001
  3. Oldham, K. B., Spanier, J., The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary, Academic Press New York (1974). (1974) Zbl0292.26011MR0361633
  4. Skula, L., Realization and GCD-Existence Theorem for generalized polynomials, In preparation. 
  5. Zariski, O., Samuel, P., Commutative Algebra, Vol. 1, D. van Nostrand Company Princeton-Toronto-New York-London (1958). (1958) Zbl0081.26501MR0090581

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