{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:67RSG42KSRJYB6UZFKLF25TTXB","short_pith_number":"pith:67RSG42K","schema_version":"1.0","canonical_sha256":"f7e323734a945380fa992a965d7673b85ac40a8f2bb0221583ef62f20c736cc3","source":{"kind":"arxiv","id":"1607.02482","version":1},"attestation_state":"computed","paper":{"title":"Polynomials Inducing the Zero Function on Local Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Cameron Wickham, Mark W. Rogers","submitted_at":"2016-07-08T18:27:04Z","abstract_excerpt":"For a Noetherian local ring (R, m) having a finite residue field of cardinality q, we study the connections between the ideal Z(R) of R[x], which is the set of polynomials that vanish on R, and the ideal Z(m), the polynomials that vanish on m, using what we call pi-polynomials: polynomials of the form p(x) = \\prod_{i = 1}^{q} (x - c_i), where c1, ..., cq is a set of representatives of the residue classes of m. When R is Henselian we prove that p(R) = m and show that a generating set for Z(R) may be obtained from a generating set for Z(m) by composing with p(x). When m is principal and has inde"},"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":"1607.02482","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-07-08T18:27:04Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"0c51fa9269ea6166219163c3c805f7877b2aaa91a0e939c703bd2a8bc2bc48f7","abstract_canon_sha256":"b05ca3e4b06331fb4c293a89450549fe23d3408d0f4647864b2ef67c967e8598"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:18.881580Z","signature_b64":"n17Ck9O6NYY4XkGh4DsIFjR3PI9+TUYJpl+C6mdcUAU3kjo8yImb5R6zqGjwTTN8HlY6Q4G2ubniGEd+XCwWBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f7e323734a945380fa992a965d7673b85ac40a8f2bb0221583ef62f20c736cc3","last_reissued_at":"2026-05-18T01:11:18.881000Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:18.881000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polynomials Inducing the Zero Function on Local Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Cameron Wickham, Mark W. Rogers","submitted_at":"2016-07-08T18:27:04Z","abstract_excerpt":"For a Noetherian local ring (R, m) having a finite residue field of cardinality q, we study the connections between the ideal Z(R) of R[x], which is the set of polynomials that vanish on R, and the ideal Z(m), the polynomials that vanish on m, using what we call pi-polynomials: polynomials of the form p(x) = \\prod_{i = 1}^{q} (x - c_i), where c1, ..., cq is a set of representatives of the residue classes of m. When R is Henselian we prove that p(R) = m and show that a generating set for Z(R) may be obtained from a generating set for Z(m) by composing with p(x). When m is principal and has inde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02482","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":"1607.02482","created_at":"2026-05-18T01:11:18.881112+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.02482v1","created_at":"2026-05-18T01:11:18.881112+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.02482","created_at":"2026-05-18T01:11:18.881112+00:00"},{"alias_kind":"pith_short_12","alias_value":"67RSG42KSRJY","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"67RSG42KSRJYB6UZ","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"67RSG42K","created_at":"2026-05-18T12:30:01.593930+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/67RSG42KSRJYB6UZFKLF25TTXB","json":"https://pith.science/pith/67RSG42KSRJYB6UZFKLF25TTXB.json","graph_json":"https://pith.science/api/pith-number/67RSG42KSRJYB6UZFKLF25TTXB/graph.json","events_json":"https://pith.science/api/pith-number/67RSG42KSRJYB6UZFKLF25TTXB/events.json","paper":"https://pith.science/paper/67RSG42K"},"agent_actions":{"view_html":"https://pith.science/pith/67RSG42KSRJYB6UZFKLF25TTXB","download_json":"https://pith.science/pith/67RSG42KSRJYB6UZFKLF25TTXB.json","view_paper":"https://pith.science/paper/67RSG42K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.02482&json=true","fetch_graph":"https://pith.science/api/pith-number/67RSG42KSRJYB6UZFKLF25TTXB/graph.json","fetch_events":"https://pith.science/api/pith-number/67RSG42KSRJYB6UZFKLF25TTXB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/67RSG42KSRJYB6UZFKLF25TTXB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/67RSG42KSRJYB6UZFKLF25TTXB/action/storage_attestation","attest_author":"https://pith.science/pith/67RSG42KSRJYB6UZFKLF25TTXB/action/author_attestation","sign_citation":"https://pith.science/pith/67RSG42KSRJYB6UZFKLF25TTXB/action/citation_signature","submit_replication":"https://pith.science/pith/67RSG42KSRJYB6UZFKLF25TTXB/action/replication_record"}},"created_at":"2026-05-18T01:11:18.881112+00:00","updated_at":"2026-05-18T01:11:18.881112+00:00"}