On polynomial equations over split octonions

TL;DR

This work tackles the problem of solving polynomial equations over the split octonion algebra $O$ with scalar coefficients and a non-scalar constant term, and applies this to computing $n$-th roots of octonions. The authors leverage the $G_2$-symmetry of $O$ to reduce constants to canonical forms and then reduce the equation to scalar-root computations on the base field, yielding a complete description of the solution set for $f(x)=c$. They prove precise counts for the number of solutions (finite, infinite, or empty) and provide explicit representations of solutions in terms of the eigenvalue data of the canonical forms, including special cases for $x^n=c$. The results advance explicit solvability in a non-associative, Cayley-Dickson setting and connect polynomial equations to $G_2$-orbit geometry, with potential implications for physics-inspired models employing split-octonion structures.

Abstract

Working over the split octonions over an algebraically closed field, we solve all polynomial equations in which all the coefficients but the constant term are scalar. As a consequence, we calculate the n-th roots of an octonion.

Theorems & Definitions (10)

• remark 1
• proposition 1: Part 1 of Proposition 3.3 from LZ_1
• definition 1
• remark 2
• lemma 1
• proof
• theorem 1
• proof
• corollary 1
• corollary 2