Roots and Dynamics of Octonion Polynomials

Adam Chapman, Solomon Vishkautsan

correctmedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves RMR(f)=R(C(f)) cleanly (Theorem 3.3) by using a trace-zero conjugator δ so that δ−1 is a central scalar multiple of δ, and then a Moufang identity to transport a root across a right scalar multiple. This fixes the nonassociativity issues and completes the reverse inclusion R(C(f))⊆RMR(f) correctly . By contrast, the model’s proof invokes the identity ((b c)μ + a c)c^{-1} = b(c μ c^{-1}) + a for a general c without justifying why it holds in a nonassociative octonion algebra; that step only works (via Moufang) when c has Tr(c)=0 so that c−1 is a central multiple of c, a hypothesis the model omits. Hence the model’s argument has a gap, while the paper’s proof is sound (see also Remark 3.1 for the companion-polynomial behavior under right multiplication) .

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The result is correct and the proof is precise about the nonassociative subtleties: the trace-zero conjugator and Moufang identity are invoked exactly where needed. The contribution consolidates the relationship between right-multiple roots and the companion polynomial, and the paper communicates the ideas succinctly and accurately.