Polynomials and degrees of maps in real normed algebras

Abstract

Abstract 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 f(t) ∈ 𝒜[t] has a root in 𝒜. As a consequence, the Jacobian determinant |J(f)| is always nonnegative in 𝒜. Moreover, using the idea of the topological degree we show that a regular polynomial g(t) 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 S is the unit sphere in 𝒜 and h 1, h 2 : S → S are smooth maps, it is shown that deg(h 1 * h 2) = deg(h 1) + deg(h 2).