| Title:
|
Polynomials and degrees of maps in real normed algebras (English) |
| Author:
|
Sakkalis, Takis |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
28 |
| Issue:
|
1 |
| Year:
|
2020 |
| Pages:
|
43-54 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal{A}$ be the algebra of quaternions $\mathbb{H}$ or octonions $\mathbb{O}$. In this manuscript an elementary proof is given, based on ideas of Cauchy and D'Alembert, of the fact that an ordinary polynomial $f(t) \in \mathcal{A} [t]$ has a root in $\mathcal{A}$. As a consequence, the Jacobian determinant $\lvert J(f)\rvert $ is always non-negative in $\mathcal{A}$. Moreover, using the idea of the topological degree we show that a regular polynomial $g(t)$ over $\mathcal{A}$ has also a root in $\mathcal{A}$. Finally, utilizing multiplication ($*$) in $\mathcal{A}$, we prove various results on the topological degree of products of maps. In particular, if $S$ is the unit sphere in $\mathcal{A}$ and $h_1, h_2\colon S \to S$ are smooth maps, it is shown that $\deg (h_1 * h_2)=\deg (h_1) + \deg (h_2)$. (English) |
| Keyword:
|
ordinary polynomials; regular polynomials; Jacobians; degrees of maps |
| MSC:
|
11R52 |
| MSC:
|
12E15 |
| MSC:
|
26B10 |
| idZBL:
|
Zbl 1470.26021 |
| idMR:
|
MR4124289 |
| . |
| Date available:
|
2020-07-22T11:50:20Z |
| Last updated:
|
2021-11-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148260 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[7] Milnor, J., Weaver, D.W.: Topology from the differentiable viewpoint.1997, Princeton University Press, MR 1487640 |
| Reference:
|
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| Reference:
|
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| . |
