{"paper":{"title":"Polynomial Equations over Octonion Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Adam Chapman","submitted_at":"2018-08-08T17:37:32Z","abstract_excerpt":"In this paper we present a complete method for finding the roots of all polynomials of the form $\\phi(z)=c_n z^n+c_{n-1} z^{n-1}+\\dots+c_1 z+c_0$ over a given octonion division algebra. When $\\phi(z)$ is monic we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of $\\phi(z)$, showing that the right eigenvalues form the conjugacy classes of the roots of $\\phi(z)$ and the left eigenvalues form a larger set than the roots of $\\phi(z)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.02875","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}