{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:7USQ2F5TBAHLDLD6D42BOB4NUW","short_pith_number":"pith:7USQ2F5T","schema_version":"1.0","canonical_sha256":"fd250d17b3080eb1ac7e1f3417078da5bda1229da83d9a3449c69cb70c60a6e0","source":{"kind":"arxiv","id":"1611.08840","version":1},"attestation_state":"computed","paper":{"title":"The Hermitian null-range of a matrix over a finite field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"E. Ballico","submitted_at":"2016-11-27T13:33:45Z","abstract_excerpt":"Let $q$ be a prime power. For $u=(u_1,\\dots ,u_n), v=(v_1,\\dots ,v_n)\\in \\mathbb {F}_{q^2}^n$ let $\\langle u,v\\rangle := \\sum _{i=1}^{n} u_i^qv_i$ be the Hermitian form of $\\mathbb {F} _{q^2}^n$. Fix an $n\\times n$ matrix $M$ over $\\mathbb {F} _{q^2}$. We study the case $k=0$ of the set $\\mathrm{Num} _k(M):= \\{\\langle u,Mu\\rangle \\mid u\\in \\mathbb {F} _{q^2}, \\langle u,u\\rangle =k\\}$. When $M$ has coefficients in $\\mathbb {F} _q$ we study the set $\\mathrm{Num} _0(M)_q:= \\{\\langle u,Mu\\rangle \\mid u\\in \\mathbb {F} _q^n\\}\\subseteq \\mathbb {F} _q$. The set $\\mathrm{Num} _1(M)$ is the numerical ra"},"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":"1611.08840","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-11-27T13:33:45Z","cross_cats_sorted":[],"title_canon_sha256":"26535e5350be3dda2452696583c5d0ae6fc663cbed8f51957cd0bde77d0d0631","abstract_canon_sha256":"ff95676395b36e4635fa54fb62e6749e049a40e6d6ab8b484bf20569c403fde5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:30.156030Z","signature_b64":"BihivB21qkk/g0xERQlSV74Qz/faBvFA5MCacUSNSV08JqEU22jz5eBh6gwqZYmnbDZeABS3OllxMtf8cOASDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd250d17b3080eb1ac7e1f3417078da5bda1229da83d9a3449c69cb70c60a6e0","last_reissued_at":"2026-05-18T00:56:30.155210Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:30.155210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Hermitian null-range of a matrix over a finite field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"E. Ballico","submitted_at":"2016-11-27T13:33:45Z","abstract_excerpt":"Let $q$ be a prime power. For $u=(u_1,\\dots ,u_n), v=(v_1,\\dots ,v_n)\\in \\mathbb {F}_{q^2}^n$ let $\\langle u,v\\rangle := \\sum _{i=1}^{n} u_i^qv_i$ be the Hermitian form of $\\mathbb {F} _{q^2}^n$. Fix an $n\\times n$ matrix $M$ over $\\mathbb {F} _{q^2}$. We study the case $k=0$ of the set $\\mathrm{Num} _k(M):= \\{\\langle u,Mu\\rangle \\mid u\\in \\mathbb {F} _{q^2}, \\langle u,u\\rangle =k\\}$. When $M$ has coefficients in $\\mathbb {F} _q$ we study the set $\\mathrm{Num} _0(M)_q:= \\{\\langle u,Mu\\rangle \\mid u\\in \\mathbb {F} _q^n\\}\\subseteq \\mathbb {F} _q$. The set $\\mathrm{Num} _1(M)$ is the numerical ra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08840","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":"1611.08840","created_at":"2026-05-18T00:56:30.155373+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.08840v1","created_at":"2026-05-18T00:56:30.155373+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08840","created_at":"2026-05-18T00:56:30.155373+00:00"},{"alias_kind":"pith_short_12","alias_value":"7USQ2F5TBAHL","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_16","alias_value":"7USQ2F5TBAHLDLD6","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_8","alias_value":"7USQ2F5T","created_at":"2026-05-18T12:30:04.600751+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/7USQ2F5TBAHLDLD6D42BOB4NUW","json":"https://pith.science/pith/7USQ2F5TBAHLDLD6D42BOB4NUW.json","graph_json":"https://pith.science/api/pith-number/7USQ2F5TBAHLDLD6D42BOB4NUW/graph.json","events_json":"https://pith.science/api/pith-number/7USQ2F5TBAHLDLD6D42BOB4NUW/events.json","paper":"https://pith.science/paper/7USQ2F5T"},"agent_actions":{"view_html":"https://pith.science/pith/7USQ2F5TBAHLDLD6D42BOB4NUW","download_json":"https://pith.science/pith/7USQ2F5TBAHLDLD6D42BOB4NUW.json","view_paper":"https://pith.science/paper/7USQ2F5T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.08840&json=true","fetch_graph":"https://pith.science/api/pith-number/7USQ2F5TBAHLDLD6D42BOB4NUW/graph.json","fetch_events":"https://pith.science/api/pith-number/7USQ2F5TBAHLDLD6D42BOB4NUW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7USQ2F5TBAHLDLD6D42BOB4NUW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7USQ2F5TBAHLDLD6D42BOB4NUW/action/storage_attestation","attest_author":"https://pith.science/pith/7USQ2F5TBAHLDLD6D42BOB4NUW/action/author_attestation","sign_citation":"https://pith.science/pith/7USQ2F5TBAHLDLD6D42BOB4NUW/action/citation_signature","submit_replication":"https://pith.science/pith/7USQ2F5TBAHLDLD6D42BOB4NUW/action/replication_record"}},"created_at":"2026-05-18T00:56:30.155373+00:00","updated_at":"2026-05-18T00:56:30.155373+00:00"}