On polynomial equations over split octonions

Alexander N. Rybalov; Artem Lopatin

On polynomial equations over split octonions

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Pith's one-line read A complete description of all solutions to scalar-coefficient polynomial equations over split octonions over algebraically closed fields, including explicit n-th root formulas.

desk verdict A complete, clean solution to scalar-coefficient polynomial equations over split octonions; the proof is sound and the only real dependency is the G2 orbit classification imported from the authors' earlier paper. read the letter →

arxiv 2412.00032 v4 pith:AO3LGP5L submitted 2024-11-21 math.RA

Artem Lopatin , Alexander N. Rybalov This is my paper · ORCID

classification math.RA

keywords equationsoctonionspolynomialsplitalgebraicallycalculateclosedcoefficients

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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Octonions are eight-dimensional numbers that do not multiply associatively. The split octonions are a version in which a nonzero number can have zero norm, which makes polynomial equations harder than in the ordinary octonions, where every nonzero element is invertible. This paper solves equations of the form f(x)=c in which f is an ordinary polynomial with scalar coefficients and the only non-scalar part is the constant c. The key tool is the automorphism group G2. It moves any octonion into one of three simple shapes: a scalar, a diagonal matrix with two different entries, or a special matrix built from a scalar and a vector u1. For the third shape, feeding it into f gives f(alpha) plus a term f'(alpha)u1, where f' is the usual derivative of f. So the whole problem reduces to solving scalar polynomial equations and checking whether certain roots are simple or multiple. The paper lists all solution orbits for each shape. If c is scalar, solutions can be a whole G2-orbit, and the number of solutions can be infinite. If c has two different eigenvalues, the solution set is a product of two root sets. If c is of the special form, the solutions are explicit and exist only for simple roots. As an application, the n-th root equation x^n=c is solved explicitly, including the case where the characteristic of the field divides n.

Extended reading notes

Core claim

Theorem 3.2: if f(ξ) is a nonzero polynomial over an algebraically closed field F without constant term and c is a split octonion, then the solution set X of f(x)=c is described completely: scalar solutions from roots f(ξ)=γ, full G2-orbits O2(ξ1,ξ2) when c=γ1O with f(ξ1)=f(ξ2)=γ, full orbits O3(ξ1) when γ is a multiple root, diagonal products of roots when c=γ1e1+γ2e2, and explicit elements ξ1 1O + (1/f'(ξ1))u1 when c=γ1O+u1 and f'(ξ1)≠0. If true, every such equation is solved modulo scalar polynomial equations, with exact cardinality bounds and n-th root formulas.

Load-bearing premise

The complete classification of G2-orbits on O stated in Proposition 2.2, the orbit criterion in Remark 2.4, and the stabilizer computation StG2(γ1e1+γ2e2)=SL3 (Lemma 2.1 of [29]) are taken as established facts from the authors' earlier paper [29]. If this classification missed an orbit or the stabilizer were larger in positive characteristic, the case analysis in Theorem 3.2 would be incomplete. This enters at Section 2.2 and is used throughout the proof of Theorem 3.2.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper solves a natural problem that was genuinely open: polynomial equations over split octonions over an algebraically closed field, where all coefficients except the constant term are scalars. The main theorem (3.2) gives a complete description of the solution set in terms of scalar roots, full G2-orbits, and explicit elements, with exact cardinality bounds in Corollary 3.3 and n-th root formulas in Corollary 3.4. That is a real step beyond Chapman's work on division octonion algebras and the linear-equation results of Lopatin and Zubkov.

What the paper does well: the proof is a clean case analysis built on Lemma 3.1 (the formula f(α1O + βu1) = f(α)1O + f'(α)βu1), plus the G2-orbit classification. I traced the proof of Theorem 3.2 and found no internal gap. The handling of the two diagonal orders in case 2 is careful, the use of the involution ℏ is legitimate, and the contradiction cases are all sound. The stabilizer claim StG2(γ1e1+γ2e2) = SL3 is used, but the stress-test note correctly observes that it is directly verifiable in every characteristic. The cardinality bounds in Corollary 3.3 are plausible and follow from the theorem.

The soft spot is exactly where the reader put it: the completeness of the whole classification rests on the G2-orbit classification from [29], which is cited but not reproved. If that classification had a missed orbit, the case analysis would be incomplete. That is a real risk, but it is bounded and not a flaw in this paper's internal logic. The paper also does not overclaim: it explicitly says the solution is modulo scalar polynomial equations, and the n-th root result is a corollary rather than the main point.

Who this is for: researchers working on octonion algebras, nonassociative polynomial equations, or G2 actions. It is a tidy, specialized result, not a breakthrough for a broad audience. I read it as a solid contribution to the algebra of split octonions, particularly useful for people who need explicit solution sets in positive characteristic. The citation pattern is honest; the self-citation to [29] is legitimate because that orbit classification is an independent, published prior result.

Recommendation: this paper deserves a serious referee. It is correct, carefully written, and fills a concrete gap. I would accept it after the usual check that the orbit classification in [29] is indeed complete.

Referee Report

0 major / 3 minor

Summary. The paper studies equations of the form f(x)=c over the split octonion algebra O over an algebraically closed field F, where f(ξ)∈F[ξ] is a non-zero polynomial without constant term and c∈O is arbitrary. Theorem 3.2 gives a complete description of the solution set X depending on the G2-orbit type of c, expressing X as a union of scalar solutions, full G2-orbits of diagonal or unipotent-type canonical elements, or explicit elements of the form ξ1 1O + f'(ξ1)^{-1}u1. Corollary 3.3 derives finiteness and cardinality bounds, and Corollary 3.4 applies the result to the n-th root equation. The proof relies on the G2-orbit classification of [29] and on a direct induction (Lemma 3.1) that evaluates f on the subspace α1O + F u1.

Significance. The paper gives a clean, complete solution to a natural class of polynomial equations over split octonions, covering arbitrary characteristic and including the isotropic case where solution sets can be infinite G2-orbits. This complements earlier work on division octonion algebras and has potential applications in the physics literature that uses split octonions. The main identity of Lemma 3.1 is elementary and correctly proved; the case analysis in Theorem 3.2 is exhaustive and internally consistent. The completeness of the result is conditional on the G2-orbit classification of [29], which is explicitly cited; the stabilizer fact used in case 2 is directly verifiable, so the residual risk lies only in the external classification. Overall this is a worthwhile contribution.

minor comments (3)

[§2.2] The sentence 'For every u, v ∈ O define δ1(u), δ2(v) from Aut(O) as follows:' is inaccurate: the displayed formulas for δ1(u) and δ2(v) use the vector operations · and × on F3, so the declaration should be 'For every u, v ∈ F3'.
[§3, proof of Theorem 3.2] In case 2(b), the relation '>' is used without definition; since the binary relation < is only required to be a dichotomy (not necessarily transitive), it would be clearer to write 'if f(ξ2) < f(ξ1)' instead of 'Let f(ξ1) > f(ξ2)'.
[Corollary 3.4] The expression 'ξ1/nγ u1' is typographically ambiguous; using a displayed fraction like \frac{ξ1}{nγ}u1 would prevent the reader from misreading it as '(ξ1/n)γ u1'.

Circularity Check

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No circularity: Theorem 3.2 reduces polynomial equations to an external G2-orbit classification, not to its own conclusion.

full rationale

The paper's central theorem (Theorem 3.2) is derived from the G2-orbit classification in Proposition 2.2 and Remark 2.4, both quoted from the authors' prior paper [29], plus the stabilizer fact StG2(γ1e1+γ2e2)=SL3 (Lemma 2.1 of [29]). Although these are self-citations and are load-bearing for the case analysis, they are not circular: [29] is an independent published classification of G2-orbits on O and on pairs of octonions, and the present paper does not use Theorem 3.2 to establish or re-derive it. The classification is characteristic-free, stated with explicit canonical representatives, and is externally checkable; it is not fitted to the equation f(x)=c. Lemma 3.1 is a direct induction computing f(α1O+βu1), and no step in the proof of Theorem 3.2 assumes the conclusion. The case 2 use of the stabilizer is a standard computation (an automorphism fixing γ1e1+γ2e2 with γ1≠γ2 fixes e1 and e2, leaving exactly the displayed SL3 action on the Zorn coordinates), and it is not derived from the equation-solving result. Therefore the derivation chain is self-contained modulo an external algebraic classification, and no claim reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the standard algebraic identities of split octonions and on the published G2-orbit classification from [29]. No free parameters are fitted, and no new entities are postulated. The orbit classification is the main external input; the rest is direct computation.

assumptions (6)

domain assumption Split octonion algebra O over algebraically closed field F is alternative, unital, 8-dimensional, with norm, trace, and involution satisfying identities (2.1)-(2.4).
Section 2.1 states these properties; all computations, including Lemma 3.1, rely on them.
standard math Artin's theorem: in an alternative algebra every subalgebra generated by two elements is associative; hence octonions are power-associative.
Introduction uses this to write a^n without parentheses.
domain assumption Proposition 2.2 of [29]: the listed three families form a minimal set of representatives of G2-orbits on O.
Used in Theorem 3.2 to normalize c and x; this is the main external input and the paper's weakest load-bearing premise.
domain assumption Remark 2.4 of [29]: orbit criterion for alpha1 1O + beta1 u1 under G2.
Used in case 3 of Theorem 3.2 to match f(xi1)1O + f'(xi1)u1 with g(gamma1O + u1).
domain assumption Lemma 2.1 of [29]: StG2(gamma1 e1 + gamma2 e2) = SL3.
Used in case 2 to determine x from the canonical diagonal representative.
standard math There exists a binary relation < on F with exactly one of alpha<beta or beta<alpha for every distinct pair.
Defined before Proposition 2.2 to choose a unique diagonal representative for each G2-orbit; no compatibility with field operations is required.

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Pith. "Pith review of On polynomial equations over split octonions." pith.science (2026). https://pith.science/paper/AO3LGP5L

@misc{pith2026241200032,
  author       = {Pith},
  title        = {Pith review of: On polynomial equations over split octonions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AO3LGP5L}},
  note         = {Machine review of arXiv:2412.00032}
}

read the original 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.

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [29]

Lopatin and A

A. Lopatin and A. N. Zubkov. Classiﬁcation of G2-orbits for pairs of octonions. Journal of Pure and Applied Algebra , 229:107875, 2025

work page 2025

[2] [1]

Bisht, S

P. Bisht, S. Dangwal, and O. Negi. Uniﬁed split octonion fo rmulation of dyons. International Journal of Theoretical Physics , 47(9):2297–2313, 2008

work page 2008

[3] [2]

C. Castro. On the noncommutative and nonassociative geo metry of octonionic space time, modiﬁed dispersion relations and grand uniﬁcation. Journal of Mathematical Physics , 48(7):073517, 2007

work page 2007

[4] [3]

B. Chanyal. Classical geometrodynamics with Zorn vector -matrix algebra for gravito-dyons. Reports on Mathematical Physics , 76(1):1–20, 2015

work page 2015

[5] [4]

B. Chanyal. Split octonion reformulation for electromag netic chiral media of massive dyons. Communications in Theoretical Physics (Beijing) , 68(6):701–710, 2017. 10 On polynomial equations over split octonions

work page 2017

[6] [5]

Chanyal, P

B. Chanyal, P. Bisht, and O. Negi. Generalized split-octon ion electrodynamics. International Journal of Theoretical Physics , 50(6):1919–1926, 2011

work page 1919

[7] [6]

Chanyal, P

B. Chanyal, P. Bisht, and O. Negi. Octonion and conservatio n laws for dyons. International Journal of Modern Physics A , 28(26):1350125, 2013

work page 2013

[8] [7]

A. Chapman. Polynomial equations over octonion algebra s. Journal of Algebra and Its Applications, 19(6):2050102, 2020

work page 2020