Polynomials and degrees of maps in real normed algebras

2018/03/11 by Sakkalis, Takis
#11R52 #12E15 #26B10 #Algebraic Geometry (math.AG) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1803.04930

Abstract

Let \calA be the algebra of quaternions ℍ or octonions \mathbbO. In this manuscript a new proof is given, based on ideas of Cauchy and D' Alembert, of the fact that an ordinary polynomial f(t) ∈ \calA [t] has a root in \calA. As a consequence, the Jacobian determinant |J(f)| is always non negative in \calA. Moreover, using the idea of the topological degree we show that a regular polynomial g(t) over \calA has also a root in \calA. Finally, utilizing multiplication (*) in \calA, we prove various results on the topological degree of products of maps. In particular, if S is the unit sphere in \calA and h₁, h₂: S → S are smooth maps, it is shown that \hboxdeg (h₁ * h₂)=\hboxdeg (h₁) + \hboxdeg (h₂).

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