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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1304.1794","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-04-05T19:49:54Z","cross_cats_sorted":[],"title_canon_sha256":"7515ba0f50dbb3630fd87cb88e111eae5c76c007465bf68318d98c0db7c543d9","abstract_canon_sha256":"dc2695b9d26ec228fbccd294392c0ecdd244308a015d90f65ac752a21991c788"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:54.660699Z","signature_b64":"XJpG7Cgdjyaz7juSJFUwiHD3VZAHeidHlp0d6DyJcI3ia7xTtZ51bnBmsDNsvB4IHEGqP4YcZIn09HCz72GYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a82c38f45ce3212055a9c51650876f12e0181b058b5e0920001592b065c57865","last_reissued_at":"2026-05-18T03:28:54.659992Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:54.659992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1304.1794","created_at":"2026-05-18T03:28:54.660116+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.1794v1","created_at":"2026-05-18T03:28:54.660116+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1794","created_at":"2026-05-18T03:28:54.660116+00:00"},{"alias_kind":"pith_short_12","alias_value":"VAWDR5C44MQS","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"VAWDR5C44MQSAVNJ","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"VAWDR5C4","created_at":"2026-05-18T12:28:04.890932+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VAWDR5C44MQSAVNJYULFBB3PCL","json":"https://pith.science/pith/VAWDR5C44MQSAVNJYULFBB3PCL.json","graph_json":"https://pith.science/api/pith-number/VAWDR5C44MQSAVNJYULFBB3PCL/graph.json","events_json":"https://pith.science/api/pith-number/VAWDR5C44MQSAVNJYULFBB3PCL/events.json","paper":"https://pith.science/paper/VAWDR5C4"},"agent_actions":{"view_html":"https://pith.science/pith/VAWDR5C44MQSAVNJYULFBB3PCL","download_json":"https://pith.science/pith/VAWDR5C44MQSAVNJYULFBB3PCL.json","view_paper":"https://pith.science/paper/VAWDR5C4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.1794&json=true","fetch_graph":"https://pith.science/api/pith-number/VAWDR5C44MQSAVNJYULFBB3PCL/graph.json","fetch_events":"https://pith.science/api/pith-number/VAWDR5C44MQSAVNJYULFBB3PCL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VAWDR5C44MQSAVNJYULFBB3PCL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VAWDR5C44MQSAVNJYULFBB3PCL/action/storage_attestation","attest_author":"https://pith.science/pith/VAWDR5C44MQSAVNJYULFBB3PCL/action/author_attestation","sign_citation":"https://pith.science/pith/VAWDR5C44MQSAVNJYULFBB3PCL/action/citation_signature","submit_replication":"https://pith.science/pith/VAWDR5C44MQSAVNJYULFBB3PCL/action/replication_record"}},"created_at":"2026-05-18T03:28:54.660116+00:00","updated_at":"2026-05-18T03:28:54.660116+00:00"}