| Title:
|
Special isomorphisms of $F[x_1,\ldots ,x_n]$ preserving GCD and their use (English) |
| Author:
|
Skula, Ladislav |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
59 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
759-771 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
polynomials in several variables over field |
| Keyword:
|
generalized polynomials in several variables over field |
| Keyword:
|
isomorphism of the ring of polynomials |
| Keyword:
|
automorphism of the ring of generalized polynomials |
| Keyword:
|
greatest common divisor of generalized polynomials |
| MSC:
|
13A05 |
| MSC:
|
13F20 |
| idZBL:
|
Zbl 1224.13024 |
| idMR:
|
MR2545654 |
| . |
| Date available:
|
2010-07-20T15:38:20Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140514 |
| . |
| Reference:
|
[1] Karásek, J., Šlapal, J.: Polynomials and Generalized Polynomials for the Theory of Control. Special Monograph.Academic Publishing House CERM Brno (2007), Czech. |
| Reference:
|
[2] Nicholson, W. K.: Introduction to Abstract Algebra.PWS-KENT Publishing Company Boston (1993). Zbl 0781.12001 |
| Reference:
|
[3] Oldham, K. B., Spanier, J.: The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary.Academic Press New York (1974). Zbl 0292.26011, MR 0361633 |
| Reference:
|
[4] Skula, L.: Realization and GCD-Existence Theorem for generalized polynomials.In preparation. |
| Reference:
|
[5] Zariski, O., Samuel, P.: Commutative Algebra, Vol. 1.D. van Nostrand Company Princeton-Toronto-New York-London (1958). Zbl 0081.26501, MR 0090581 |
| . |
