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Sakkalis, Takis
Communications in Mathematics, 06/2020, Letnik: 28, Številka: 1Journal Article
Let 𝒜 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 ) ∈ 𝒜 has a root in 𝒜. As a consequence, the Jacobian determinant | )| is always nonnegative in 𝒜. Moreover, using the idea of the topological degree we show that a regular polynomial ) over 𝒜 has also a root in 𝒜. Finally, utilizing multiplication (*) in 𝒜, we prove various results on the topological degree of products of maps. In particular, if is the unit sphere in 𝒜 and , : are smooth maps, it is shown that deg( * ) = deg( ) + deg( ).
