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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
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:
[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 0781.12001
[3] Oldham, K. B., Spanier, J.: The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary. Academic Press New York (1974). MR 0361633 | Zbl 0292.26011
[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 0090581 | Zbl 0081.26501
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