Polynomials and degrees of maps in real normed algebras

Abstract

Let $\cal{A}$ be the algebra of quaternions $\mathbb{H}$ or octonions $\mathbb{O}$. In this manuscript a new proof is given, based on ideas of Cauchy and D' Alembert, of the fact that an ordinary polynomial $f(t) \in {\cal{A}}\, [t]$ has a root in $\cal{A}$. As a consequence, the Jacobian determinant $|J(f)|$ is always non negative in $\cal{A}$. Moreover, using the idea of the topological degree we show that a regular polynomial $g(t)$ over $\cal{A}$ has also a root in $\cal{A}$. Finally, utilizing multiplication $(*)$ in $\cal{A}$, we prove various results on the topological degree of products of maps. In particular, if $S$ is the unit sphere in $\cal{A}$ and $h_1, h_2: S \to S$ are smooth maps, it is shown that $\hbox{deg} (h_1 * h_2)=\hbox{deg} (h_1) + \hbox{deg} (h_2)$.