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Let $\\vec{x}=(x_1,x_2,\\ldots,x_{d},0,0,\\ldots,0)\\in\\mathbb{F}^{n}$, with $n-d$ zeros at the end. Let $\\vec{R}: \\mathbb{F}^n \\to\\mathbb{F}^n$ be the cyclic shift operator to the right, e.g. $\\vec{R}\\,\\vec{x} = (0,x_1,x_2,\\ldots,x_{d},0,0,\\ldots,0)$. Is there a vector $\\vec{x} \\in \\mathbb{F}^n$, such that the $n-d+1$ vectors $\\vec{x},\\vec{R}\\vec{x}, \\ldots ,\\vec{R}^{n-d}\\vec{x}$ complete the set $"},"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":"1905.11812","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2019-05-27T17:35:26Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"74a61a27f086112bab5bf6c3f2f25e70ba905da0e284651ebaab99e089cbcdd8","abstract_canon_sha256":"ffd0ba166caedb8317de11bfba191283571096e701bc707c76d579274dc0a401"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:50.185405Z","signature_b64":"707I7252ILHJ0vPdprBvg8NX+DiNKdc1OptKzkqnhayiNfDIZo+FhUX0qhFTdywtNfnprupgS06X6OffwlhlAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1068ac5ff0418a2c567b09c15badea6c445d6066f56049550eed1ff6d3210e07","last_reissued_at":"2026-05-17T23:44:50.184695Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:50.184695Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On completion of a linearly independent set to a basis with shifts of a fixed vector","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.RA","authors_text":"Marek Rychlik","submitted_at":"2019-05-27T17:35:26Z","abstract_excerpt":"Let $\\mathbb{F}$ be an infinite field. Let $n$ be a positive integer and let $1\\leq d\\leq n$. Let $\\vec{f}_1, \\vec{f}_2, \\ldots, \\vec{f}_{d-1} \\in \\mathbb{F}^{n}$ be $d-1$ linearly independent vectors. Let $\\vec{x}=(x_1,x_2,\\ldots,x_{d},0,0,\\ldots,0)\\in\\mathbb{F}^{n}$, with $n-d$ zeros at the end. Let $\\vec{R}: \\mathbb{F}^n \\to\\mathbb{F}^n$ be the cyclic shift operator to the right, e.g. $\\vec{R}\\,\\vec{x} = (0,x_1,x_2,\\ldots,x_{d},0,0,\\ldots,0)$. Is there a vector $\\vec{x} \\in \\mathbb{F}^n$, such that the $n-d+1$ vectors $\\vec{x},\\vec{R}\\vec{x}, \\ldots ,\\vec{R}^{n-d}\\vec{x}$ complete the set $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.11812","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":"1905.11812","created_at":"2026-05-17T23:44:50.184805+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.11812v1","created_at":"2026-05-17T23:44:50.184805+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.11812","created_at":"2026-05-17T23:44:50.184805+00:00"},{"alias_kind":"pith_short_12","alias_value":"CBUKYX7QIGFC","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"CBUKYX7QIGFCYVT3","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"CBUKYX7Q","created_at":"2026-05-18T12:33:12.712433+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/CBUKYX7QIGFCYVT3BHAVXLPKNR","json":"https://pith.science/pith/CBUKYX7QIGFCYVT3BHAVXLPKNR.json","graph_json":"https://pith.science/api/pith-number/CBUKYX7QIGFCYVT3BHAVXLPKNR/graph.json","events_json":"https://pith.science/api/pith-number/CBUKYX7QIGFCYVT3BHAVXLPKNR/events.json","paper":"https://pith.science/paper/CBUKYX7Q"},"agent_actions":{"view_html":"https://pith.science/pith/CBUKYX7QIGFCYVT3BHAVXLPKNR","download_json":"https://pith.science/pith/CBUKYX7QIGFCYVT3BHAVXLPKNR.json","view_paper":"https://pith.science/paper/CBUKYX7Q","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.11812&json=true","fetch_graph":"https://pith.science/api/pith-number/CBUKYX7QIGFCYVT3BHAVXLPKNR/graph.json","fetch_events":"https://pith.science/api/pith-number/CBUKYX7QIGFCYVT3BHAVXLPKNR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CBUKYX7QIGFCYVT3BHAVXLPKNR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CBUKYX7QIGFCYVT3BHAVXLPKNR/action/storage_attestation","attest_author":"https://pith.science/pith/CBUKYX7QIGFCYVT3BHAVXLPKNR/action/author_attestation","sign_citation":"https://pith.science/pith/CBUKYX7QIGFCYVT3BHAVXLPKNR/action/citation_signature","submit_replication":"https://pith.science/pith/CBUKYX7QIGFCYVT3BHAVXLPKNR/action/replication_record"}},"created_at":"2026-05-17T23:44:50.184805+00:00","updated_at":"2026-05-17T23:44:50.184805+00:00"}