Roots and Dynamics of Octonion Polynomials

TL;DR

This paper investigates the roots of octonion polynomials, including right and left scalar multiples, and explores the dynamics of quadratic real octonion polynomials, classifying fixed points and analyzing pseudo-periodic behavior.

Contribution

It provides new results on roots of octonion polynomials and classifies the dynamics of quadratic octonion polynomial maps, extending understanding of their algebraic and dynamical properties.

Findings

Roots of right scalar multiples are characterized.

Roots of left scalar multiples are discussed over fields of characteristic not 2.

Fixed points of quadratic octonion polynomials are classified as attracting, repelling, or ambivalent.

Abstract

This paper is devoted to several new results concerning (standard) octonion polynomials. The first is the determination of the roots of all right scalar multiples of octonion polynomials. The roots of left multiples are also discussed, especially over fields of characteristic not 2. We then turn to study the dynamics of monic quadratic real octonion polynomials, classifying the fixed points into attracting, repelling and ambivalent, and concluding with a discussion on the behavior of pseudo-periodic points.