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PreprintCC BY-SA 4.0arXiv

General Polynomials and Eigenvalues Over Cayley--Dickson Algebras

  • Adam Chapman
  • Ilan Levin
  • Solomon Vishkautsan
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Abstract

In this paper, we provide an explicit method for determining the zero set of monic quadratic general polynomials over Cayley--Dickson algebras with any base field of characteristic not equal to $2$, which reduces the problem to finding the roots of univariate polynomials over the base field. For the special case of locally-complex Cayley--Dickson algebras over the reals, we prove that every such polynomial has a root. We prove the existence of right eigenvalues for any 2 × 2 matrix over Cayley's octonions (the real octonion division algebra), and our root-finding method allows the computation of some of the right eigenvalues.

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Cite

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BibTeX
@misc{chapman2026general,
  title = {{General Polynomials and Eigenvalues Over Cayley--Dickson Algebras}},
  author = {Adam Chapman and Ilan Levin and Solomon Vishkautsan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.23613},
  eprint = {2607.23613},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.23613},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - General Polynomials and Eigenvalues Over Cayley--Dickson Algebras
AU  - Adam Chapman
AU  - Ilan Levin
AU  - Solomon Vishkautsan
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2607.23613
UR  - https://arxiv.org/abs/2607.23613
AB  - In this paper, we provide an explicit method for determining the zero set of monic quadratic general polynomials over Cayley--Dickson algebras with any base field of characteristic not equal to $2$, which reduces the problem to finding the roots of univariate polynomials over the base field. For the special case of locally-complex Cayley--Dickson algebras over the reals, we prove that every such polynomial has a root. We prove the existence of right eigenvalues for any 2 × 2 matrix over Cayley's octonions (the real octonion division algebra), and our root-finding method allows the computation of some of the right eigenvalues.
ER  -