{"paper":{"title":"Minimal height companion matrices for Euclid polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Eunice Y. S. Chan, Robert M. Corless","submitted_at":"2017-12-12T17:46:08Z","abstract_excerpt":"We define Euclid polynomials $E_{k+1}(\\lambda) = E_{k}(\\lambda)\\left(E_{k}(\\lambda) - 1\\right) + 1$ and $E_{1}(\\lambda) = \\lambda + 1$ in analogy to Euclid numbers $e_k = E_{k}(1)$. We show how to construct companion matrices $\\mathbb{E}_k$, so $E_k(\\lambda) = \\operatorname{det}\\left(\\lambda\\mathbf{I} - \\mathbb{E}_{k}\\right)$, of height 1 (and thus of minimal height over all integer companion matrices for $E_{k}(\\lambda)$). We prove various properties of these objects, and give experimental confirmation of some unproved properties."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04405","kind":"arxiv","version":1},"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"}