{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:UWFH5KLSKAYKB6CQ6SPRHJQ3LG","short_pith_number":"pith:UWFH5KLS","schema_version":"1.0","canonical_sha256":"a58a7ea9725030a0f850f49f13a61b598afc8351f839d1a04ce8c35fb597b806","source":{"kind":"arxiv","id":"1712.00387","version":1},"attestation_state":"computed","paper":{"title":"Footprint and minimum distance functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.AG","math.CO","math.IT"],"primary_cat":"math.AC","authors_text":"Luis N\\'u\\~nez-Betancourt, Rafael H. Villarreal, Yuriko Pitones","submitted_at":"2017-12-01T16:09:28Z","abstract_excerpt":"Let $S$ be a polynomial ring over a field $K$, with a monomial order $\\prec$, and let $I$ be an unmixed graded ideal of $S$. In this paper we study two functions associated to $I$: the minimum distance function $\\delta_I$ and the footprint function ${\\rm fp}_I$. It is shown that $\\delta_I$ is positive and that ${\\rm fp}_I$ is positive if the initial ideal of $I$ is unmixed. Then we show that if $I$ is radical and its associated primes are generated by linear forms, then $\\delta_I$ is strictly decreasing until it reaches the asymptotic value $1$. If $I$ is the edge ideal of a Cohen--Macaulay bi"},"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":"1712.00387","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-12-01T16:09:28Z","cross_cats_sorted":["cs.IT","math.AG","math.CO","math.IT"],"title_canon_sha256":"9640f8954909651c8cab58699c8447c53d25455fcbbe4c2d7b7e5e7cc9069e03","abstract_canon_sha256":"d465aec3599c9fcd2c547dd47ef2ea1a149752f8d6a987ef289ae55cad0581a2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:04.801991Z","signature_b64":"SafKQyuUtia1hSk4SY8qhISdd1dd3zUgG3CDp81MW5IYNhJKvtTnJGIrNcsD+F5v7+8LaAQDOUHhxvxoyfvmBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a58a7ea9725030a0f850f49f13a61b598afc8351f839d1a04ce8c35fb597b806","last_reissued_at":"2026-05-17T23:44:04.801523Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:04.801523Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Footprint and minimum distance functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.AG","math.CO","math.IT"],"primary_cat":"math.AC","authors_text":"Luis N\\'u\\~nez-Betancourt, Rafael H. Villarreal, Yuriko Pitones","submitted_at":"2017-12-01T16:09:28Z","abstract_excerpt":"Let $S$ be a polynomial ring over a field $K$, with a monomial order $\\prec$, and let $I$ be an unmixed graded ideal of $S$. In this paper we study two functions associated to $I$: the minimum distance function $\\delta_I$ and the footprint function ${\\rm fp}_I$. It is shown that $\\delta_I$ is positive and that ${\\rm fp}_I$ is positive if the initial ideal of $I$ is unmixed. Then we show that if $I$ is radical and its associated primes are generated by linear forms, then $\\delta_I$ is strictly decreasing until it reaches the asymptotic value $1$. If $I$ is the edge ideal of a Cohen--Macaulay bi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.00387","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":"1712.00387","created_at":"2026-05-17T23:44:04.801605+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.00387v1","created_at":"2026-05-17T23:44:04.801605+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.00387","created_at":"2026-05-17T23:44:04.801605+00:00"},{"alias_kind":"pith_short_12","alias_value":"UWFH5KLSKAYK","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_16","alias_value":"UWFH5KLSKAYKB6CQ","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_8","alias_value":"UWFH5KLS","created_at":"2026-05-18T12:31:49.984773+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/UWFH5KLSKAYKB6CQ6SPRHJQ3LG","json":"https://pith.science/pith/UWFH5KLSKAYKB6CQ6SPRHJQ3LG.json","graph_json":"https://pith.science/api/pith-number/UWFH5KLSKAYKB6CQ6SPRHJQ3LG/graph.json","events_json":"https://pith.science/api/pith-number/UWFH5KLSKAYKB6CQ6SPRHJQ3LG/events.json","paper":"https://pith.science/paper/UWFH5KLS"},"agent_actions":{"view_html":"https://pith.science/pith/UWFH5KLSKAYKB6CQ6SPRHJQ3LG","download_json":"https://pith.science/pith/UWFH5KLSKAYKB6CQ6SPRHJQ3LG.json","view_paper":"https://pith.science/paper/UWFH5KLS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.00387&json=true","fetch_graph":"https://pith.science/api/pith-number/UWFH5KLSKAYKB6CQ6SPRHJQ3LG/graph.json","fetch_events":"https://pith.science/api/pith-number/UWFH5KLSKAYKB6CQ6SPRHJQ3LG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UWFH5KLSKAYKB6CQ6SPRHJQ3LG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UWFH5KLSKAYKB6CQ6SPRHJQ3LG/action/storage_attestation","attest_author":"https://pith.science/pith/UWFH5KLSKAYKB6CQ6SPRHJQ3LG/action/author_attestation","sign_citation":"https://pith.science/pith/UWFH5KLSKAYKB6CQ6SPRHJQ3LG/action/citation_signature","submit_replication":"https://pith.science/pith/UWFH5KLSKAYKB6CQ6SPRHJQ3LG/action/replication_record"}},"created_at":"2026-05-17T23:44:04.801605+00:00","updated_at":"2026-05-17T23:44:04.801605+00:00"}