Geodesic
Polynomials and degrees of maps in real normed algebras
Communications in Mathematics, Tome 28 (2020) no. 1, pp. 43-54 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\deg (h_1 * h_2)=\deg (h_1) + \deg (h_2)$.
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 $\deg (h_1 * h_2)=\deg (h_1) + \deg (h_2)$.
Classification : 11R52, 12E15, 26B10
Keywords: ordinary polynomials; regular polynomials; Jacobians; degrees of maps 
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Sakkalis, Takis. Polynomials and degrees of maps in real normed algebras. Communications in Mathematics, Tome 28 (2020) no. 1, pp. 43-54. http://geodesic.mathdoc.fr/item/COMIM_2020_28_1_a3/

[1] Baez, J.C.: The octonions. Bull. Amer. Math. Soc., 39, 2002, 145-205, | DOI | MR

[2] Bourbaki, N.: General Topology, Part 2. 1966, Hermann, Paris, | MR

[3] Eilenberg, S., Niven, I.: The ``fundamental theorem of algebra'' for quaternions. Bull. Amer. Math. Soc., 50, 4, 1944, 246-248, | DOI | MR

[4] Gentili, G., Struppa, D.C.: On the multiplicity of zeros of polynomials with quaternionic coefficients. Milan J. Math., 76, 1, 2008, 15-25, | DOI | MR

[5] Gentili, G., Struppa, D.C., Vlacci, F.: The fundamental theorem of algebra for Hamilton and Cayley numbers. Mathematische Zeitschrift, 259, 4, 2008, 895-902, Springer, | DOI | MR

[6] Gordon, B., Motzkin, T.S.: On the zeros of polynomials over division rings. Transactions of the American Mathematical Society, 116, 1965, 218-226, JSTOR, | DOI | MR

[7] Milnor, J., Weaver, D.W.: Topology from the differentiable viewpoint. 1997, Princeton University Press, | MR

[8] Rodríguez-Ordóñez, H.: A note on the fundamental theorem of algebra for the octonions. Expositiones Mathematicae, 25, 4, 2007, 355-361, Elsevier, | DOI | MR

[9] Topuridze, N.: On the roots of polynomials over division algebras. Georgian Math. Journal, 10, 4, 2003, 745-762, Walter de Gruyter, | DOI | MR