Previous |  Up |  Next


ordinary polynomials; regular polynomials; Jacobians; degrees of maps
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 $\deg (h_1 * h_2)=\deg (h_1) + \deg (h_2)$.
[1] Baez, J.C.: The octonions. Bull. Amer. Math. Soc., 39, 2002, 145-205, DOI 10.1090/S0273-0979-01-00934-X | MR 1886087
[2] Bourbaki, N.: General Topology, Part 2. 1966, Hermann, Paris, MR 0141067
[3] Eilenberg, S., Niven, I.: The ``fundamental theorem of algebra'' for quaternions. Bull. Amer. Math. Soc., 50, 4, 1944, 246-248, DOI 10.1090/S0002-9904-1944-08125-1 | MR 0009588
[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 10.1007/s00032-008-0093-0 | MR 2465984
[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 10.1007/s00209-007-0254-9 | MR 2403747
[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 10.1090/S0002-9947-1965-0195853-2 | MR 0195853
[7] Milnor, J., Weaver, D.W.: Topology from the differentiable viewpoint. 1997, Princeton University Press, MR 1487640
[8] Rodríguez-Ordóñez, H.: A note on the fundamental theorem of algebra for the octonions. Expositiones Mathematicae, 25, 4, 2007, 355-361, Elsevier, DOI 10.1016/j.exmath.2007.02.005 | MR 2360922
[9] Topuridze, N.: On the roots of polynomials over division algebras. Georgian Math. Journal, 10, 4, 2003, 745-762, Walter de Gruyter, DOI 10.1515/GMJ.2003.745 | MR 2037774
Partner of
EuDML logo