{"paper":{"title":"Is every matrix similar to a polynomial in a companion matrix?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Fernando Szechtman, Natalio H. Guersenzvaig","submitted_at":"2013-04-05T19:49:54Z","abstract_excerpt":"Given a field $F$, an integer $n\\geq 1$, and a matrix $A\\in M_n(F)$, are there polynomials $f,g\\in F[X]$, with $f$ monic of degree $n$, such that $A$ is similar to $g(C_f)$, where $C_f$ is the companion matrix of $f$? For infinite fields the answer is easily seen to positive, so we concentrate on finite fields. In this case we give an affirmative answer, provided $|F|\\geq n-2$. Moreover, for any finite field $F$, with $|F|=m$, we construct a matrix $A\\in M_{m+3}(F)$ that is not similar to any matrix of the form $g(C_f)$.\n  Of use above, but also of independent interest, is a constructive proce"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1794","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"}