Geodesic
Special isomorphisms of $F[x_1,\ldots ,x_n]$ preserving GCD and their use
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 759-771

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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.
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.
Classification : 13A05, 13F20
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 
Skula, Ladislav. Special isomorphisms of $F[x_1,\ldots ,x_n]$ preserving GCD and their use. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 759-771. http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a14/
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     title = {Special isomorphisms of $F[x_1,\ldots ,x_n]$ preserving {GCD} and their use},
     journal = {Czechoslovak Mathematical Journal},
     pages = {759--771},
     year = {2009},
     volume = {59},
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     mrnumber = {2545654},
     zbl = {1224.13024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a14/}
}
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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.

[2] Nicholson, W. K.: Introduction to Abstract Algebra. PWS-KENT Publishing Company Boston (1993). | Zbl

[3] Oldham, K. B., Spanier, J.: The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary. Academic Press New York (1974). | MR | Zbl

[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). | MR | Zbl