Polynomials and degrees of maps in real normed algebras

Sakkalis, Takis. "Polynomials and degrees of maps in real normed algebras." Communications in Mathematics 28.1 (2020): 43-54. <http://eudml.org/doc/297148>.

@article{Sakkalis2020,
abstract = {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 $\vert J(f)\vert $ 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 $h_1, h_2\colon S \rightarrow S$ are smooth maps, it is shown that $\deg (h_1 * h_2)=\deg (h_1) + \deg (h_2)$.},
author = {Sakkalis, Takis},
journal = {Communications in Mathematics},
keywords = {ordinary polynomials; regular polynomials; Jacobians; degrees of maps},
language = {eng},
number = {1},
pages = {43-54},
publisher = {University of Ostrava},
title = {Polynomials and degrees of maps in real normed algebras},
url = {http://eudml.org/doc/297148},
volume = {28},
year = {2020},
}

TY - JOUR
AU - Sakkalis, Takis
TI - Polynomials and degrees of maps in real normed algebras
JO - Communications in Mathematics
PY - 2020
PB - University of Ostrava
VL - 28
IS - 1
SP - 43
EP - 54
AB - 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 $\vert J(f)\vert $ 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 $h_1, h_2\colon S \rightarrow S$ are smooth maps, it is shown that $\deg (h_1 * h_2)=\deg (h_1) + \deg (h_2)$.
LA - eng
KW - ordinary polynomials; regular polynomials; Jacobians; degrees of maps
UR - http://eudml.org/doc/297148
ER -