{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:FJDFMUD33HIQZLK5UGIZU5KH3S","short_pith_number":"pith:FJDFMUD3","schema_version":"1.0","canonical_sha256":"2a4656507bd9d10cad5da1919a7547dcbcc79a4696769cb8abe1e9239a8ac0ae","source":{"kind":"arxiv","id":"2606.24349","version":1},"attestation_state":"computed","paper":{"title":"Sharp bounds for minimal dependencies of linear-form powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heng Li, Xizhi Liu","submitted_at":"2026-06-23T09:40:15Z","abstract_excerpt":"Motivated by the dimension-bound part of a problem of Bukh, we study Veronese circuits: how large can the span of $t$ linear forms be if their $m$-th powers are minimally linearly dependent? We prove the sharp finite dimension bound \\[\n  \\dim L\\leq \\frac{t+m-2}{m}. \\] Here $\\ell_1,\\ldots,\\ell_t$ are nonzero homogeneous linear forms over a field of characteristic zero, the powers $\\ell_1^m,\\ldots,\\ell_t^m$ form a circuit, and $L=\\Span\\{\\ell_1,\\ldots,\\ell_t\\}$. Rational-normal-curve configurations attain equality for infinitely many pairs $(t,\\dim L)$; in particular, the affine bound itself is s"},"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":"2606.24349","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T09:40:15Z","cross_cats_sorted":[],"title_canon_sha256":"89a11175079dad956dcf028a95632f1df511a861dc54751ac620eb8469a4bf3e","abstract_canon_sha256":"242e9bb6760db4dabbbbba33253d66c9b90c71ce354464a351610fe24b812aa8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:15:27.946988Z","signature_b64":"T/BFZqz0grMKOppxLAP3TJBK0x6bcqfLzLvCzCq/jnTcyG8IIZR2IlMMnD7/oekeVfm1Kk7SErbJmKGxBHA+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a4656507bd9d10cad5da1919a7547dcbcc79a4696769cb8abe1e9239a8ac0ae","last_reissued_at":"2026-06-24T01:15:27.946467Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:15:27.946467Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp bounds for minimal dependencies of linear-form powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heng Li, Xizhi Liu","submitted_at":"2026-06-23T09:40:15Z","abstract_excerpt":"Motivated by the dimension-bound part of a problem of Bukh, we study Veronese circuits: how large can the span of $t$ linear forms be if their $m$-th powers are minimally linearly dependent? We prove the sharp finite dimension bound \\[\n  \\dim L\\leq \\frac{t+m-2}{m}. \\] Here $\\ell_1,\\ldots,\\ell_t$ are nonzero homogeneous linear forms over a field of characteristic zero, the powers $\\ell_1^m,\\ldots,\\ell_t^m$ form a circuit, and $L=\\Span\\{\\ell_1,\\ldots,\\ell_t\\}$. Rational-normal-curve configurations attain equality for infinitely many pairs $(t,\\dim L)$; in particular, the affine bound itself is s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24349","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.24349/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2606.24349","created_at":"2026-06-24T01:15:27.946525+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.24349v1","created_at":"2026-06-24T01:15:27.946525+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24349","created_at":"2026-06-24T01:15:27.946525+00:00"},{"alias_kind":"pith_short_12","alias_value":"FJDFMUD33HIQ","created_at":"2026-06-24T01:15:27.946525+00:00"},{"alias_kind":"pith_short_16","alias_value":"FJDFMUD33HIQZLK5","created_at":"2026-06-24T01:15:27.946525+00:00"},{"alias_kind":"pith_short_8","alias_value":"FJDFMUD3","created_at":"2026-06-24T01:15:27.946525+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/FJDFMUD33HIQZLK5UGIZU5KH3S","json":"https://pith.science/pith/FJDFMUD33HIQZLK5UGIZU5KH3S.json","graph_json":"https://pith.science/api/pith-number/FJDFMUD33HIQZLK5UGIZU5KH3S/graph.json","events_json":"https://pith.science/api/pith-number/FJDFMUD33HIQZLK5UGIZU5KH3S/events.json","paper":"https://pith.science/paper/FJDFMUD3"},"agent_actions":{"view_html":"https://pith.science/pith/FJDFMUD33HIQZLK5UGIZU5KH3S","download_json":"https://pith.science/pith/FJDFMUD33HIQZLK5UGIZU5KH3S.json","view_paper":"https://pith.science/paper/FJDFMUD3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.24349&json=true","fetch_graph":"https://pith.science/api/pith-number/FJDFMUD33HIQZLK5UGIZU5KH3S/graph.json","fetch_events":"https://pith.science/api/pith-number/FJDFMUD33HIQZLK5UGIZU5KH3S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FJDFMUD33HIQZLK5UGIZU5KH3S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FJDFMUD33HIQZLK5UGIZU5KH3S/action/storage_attestation","attest_author":"https://pith.science/pith/FJDFMUD33HIQZLK5UGIZU5KH3S/action/author_attestation","sign_citation":"https://pith.science/pith/FJDFMUD33HIQZLK5UGIZU5KH3S/action/citation_signature","submit_replication":"https://pith.science/pith/FJDFMUD33HIQZLK5UGIZU5KH3S/action/replication_record"}},"created_at":"2026-06-24T01:15:27.946525+00:00","updated_at":"2026-06-24T01:15:27.946525+00:00"}