{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:GPKPDC6EAOTLJXDO67UOWVNDJM","short_pith_number":"pith:GPKPDC6E","schema_version":"1.0","canonical_sha256":"33d4f18bc403a6b4dc6ef7e8eb55a34b3bb6c9fa55e6d7c621e27fb5657d7a90","source":{"kind":"arxiv","id":"1110.2822","version":1},"attestation_state":"computed","paper":{"title":"The Weak Lefschetz Property for monomial complete intersections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Adela Vraciu, Andrew R. Kustin","submitted_at":"2011-10-13T00:57:32Z","abstract_excerpt":"Let $A=\\pmb k[x_1,...,x_n]/{(x_1^d,...,x_n^d)}$, where $\\pmb k$ is an infinite field. If $\\pmb k$ has characteristic zero, then Stanley proved that $A$ has the Weak Lefschetz Property (WLP). Henceforth, $\\pmb k$ has positive characteristic $p$. If $n=3$, then Brenner and Kaid have identified all $d$, as a function of $p$, for which $A$ has the WLP. In the present paper, the analogous project is carried out for $4\\le n$. If $4\\le n$ and $p=2$, then $A$ has the WLP if and only if $d=1$. If $n=4$ and $p$ is odd, then we prove that $A$ has the WLP if and only if $d=kq+r$ for integers $k,q,d$ with "},"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":"1110.2822","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-10-13T00:57:32Z","cross_cats_sorted":[],"title_canon_sha256":"9473b78f38be72b792aa8345201d3ff2146d871c02b60e4316e6ad74326c25bc","abstract_canon_sha256":"f89760aac1633044c60ea98254e15e30cac9ec30073916e97ee035caf5106042"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:06.904677Z","signature_b64":"iDjCmzcgO2wO/zS5tLek4zeYbspeV5V1lGe8uRKBLfui0yCwcvL0F/fqnK5sMcHqYNwlh0+vgf7pJzhY3ElLBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33d4f18bc403a6b4dc6ef7e8eb55a34b3bb6c9fa55e6d7c621e27fb5657d7a90","last_reissued_at":"2026-05-18T04:11:06.904164Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:06.904164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Weak Lefschetz Property for monomial complete intersections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Adela Vraciu, Andrew R. Kustin","submitted_at":"2011-10-13T00:57:32Z","abstract_excerpt":"Let $A=\\pmb k[x_1,...,x_n]/{(x_1^d,...,x_n^d)}$, where $\\pmb k$ is an infinite field. If $\\pmb k$ has characteristic zero, then Stanley proved that $A$ has the Weak Lefschetz Property (WLP). Henceforth, $\\pmb k$ has positive characteristic $p$. If $n=3$, then Brenner and Kaid have identified all $d$, as a function of $p$, for which $A$ has the WLP. In the present paper, the analogous project is carried out for $4\\le n$. If $4\\le n$ and $p=2$, then $A$ has the WLP if and only if $d=1$. If $n=4$ and $p$ is odd, then we prove that $A$ has the WLP if and only if $d=kq+r$ for integers $k,q,d$ with "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2822","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":"1110.2822","created_at":"2026-05-18T04:11:06.904236+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.2822v1","created_at":"2026-05-18T04:11:06.904236+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.2822","created_at":"2026-05-18T04:11:06.904236+00:00"},{"alias_kind":"pith_short_12","alias_value":"GPKPDC6EAOTL","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"GPKPDC6EAOTLJXDO","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"GPKPDC6E","created_at":"2026-05-18T12:26:30.835961+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/GPKPDC6EAOTLJXDO67UOWVNDJM","json":"https://pith.science/pith/GPKPDC6EAOTLJXDO67UOWVNDJM.json","graph_json":"https://pith.science/api/pith-number/GPKPDC6EAOTLJXDO67UOWVNDJM/graph.json","events_json":"https://pith.science/api/pith-number/GPKPDC6EAOTLJXDO67UOWVNDJM/events.json","paper":"https://pith.science/paper/GPKPDC6E"},"agent_actions":{"view_html":"https://pith.science/pith/GPKPDC6EAOTLJXDO67UOWVNDJM","download_json":"https://pith.science/pith/GPKPDC6EAOTLJXDO67UOWVNDJM.json","view_paper":"https://pith.science/paper/GPKPDC6E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.2822&json=true","fetch_graph":"https://pith.science/api/pith-number/GPKPDC6EAOTLJXDO67UOWVNDJM/graph.json","fetch_events":"https://pith.science/api/pith-number/GPKPDC6EAOTLJXDO67UOWVNDJM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GPKPDC6EAOTLJXDO67UOWVNDJM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GPKPDC6EAOTLJXDO67UOWVNDJM/action/storage_attestation","attest_author":"https://pith.science/pith/GPKPDC6EAOTLJXDO67UOWVNDJM/action/author_attestation","sign_citation":"https://pith.science/pith/GPKPDC6EAOTLJXDO67UOWVNDJM/action/citation_signature","submit_replication":"https://pith.science/pith/GPKPDC6EAOTLJXDO67UOWVNDJM/action/replication_record"}},"created_at":"2026-05-18T04:11:06.904236+00:00","updated_at":"2026-05-18T04:11:06.904236+00:00"}