Polynomials and degrees of maps in real normed algebras

TL;DR

This paper proves that polynomials over quaternions and octonions have roots, explores the non-negativity of the Jacobian determinant, and examines the topological degree of maps and their products in these algebras.

Contribution

It provides new proofs of root existence for polynomials in real normed algebras and analyzes the topological degree of map products using algebraic multiplication.

Findings

Polynomials over quaternions and octonions always have roots.

The Jacobian determinant of such polynomials is non-negative.

The degree of the product of two maps equals the sum of their degrees.

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 $\hbox{deg} (h_1 * h_2)=\hbox{deg} (h_1) + \hbox{deg} (h_2)$.