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  • Author: Takis Sakkalis x
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Abstract

Let π’œ be the algebra of quaternions ℍ or octonions 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) ∈ π’œ[t] has a root in π’œ. As a consequence, the Jacobian determinant |J(f)| is always nonnegative in π’œ. Moreover, using the idea of the topological degree we show that a regular polynomial g(t) over π’œ has also a root in π’œ. Finally, utilizing multiplication (*) in π’œ, we prove various results on the topological degree of products of maps. In particular, if S is the unit sphere in π’œ and h 1, h 2 : S β†’ S are smooth maps, it is shown that deg(h 1 * h 2) = deg(h 1) + deg(h 2).