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Assuming that the grade of the ideal generated by the $k$-minors of $M$ is at least $m - k + 2$ for $1 \\leq \\forall k \\leq m$, we will study the associated primes of $I^n$ for $\\forall n > 0$. Moreover, we compute the saturation of $I^n$ for $1 \\leq \\forall n \\leq m$ in the case where $R$ is a Cohen-Macaulay ring and the entries of $M$ are powers of elements that form an sop for $R$."},"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":"1307.0593","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-07-02T06:12:26Z","cross_cats_sorted":[],"title_canon_sha256":"4deb4f7677b9f47bcc61c0c88cf25eded233b73fe49bc4b311cf6f3c1e92ff94","abstract_canon_sha256":"91e213c81b9b956e951cade14e44d9aadb9bd204a0976d3b792c07893425fc87"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:25.207719Z","signature_b64":"OdfpjdAdgpBn+K/SPP6TvgFqVy64EdT3HwZoJxbU+GWyGE0Qfnsdbl54dmDAOFnbLLYV/tURs5sRVqt0sRF0Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a9d20a3a02999f5679e8e0e30b0b590c09c7629a74d4f30e57799a97ed8865e","last_reissued_at":"2026-05-18T03:19:25.206971Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:25.206971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Saturations of powers of certain determinantal ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Koji Nishida, Kosuke Fukumuro, Taro Inagawa","submitted_at":"2013-07-02T06:12:26Z","abstract_excerpt":"Let $R$ be a Noetherian local ring and $m$ a positive integer. Let $I$ be the ideal of $R$ generated by the maximal minors of an $m \\times (m + 1)$ matrix $M$ with entries in $R$. Assuming that the grade of the ideal generated by the $k$-minors of $M$ is at least $m - k + 2$ for $1 \\leq \\forall k \\leq m$, we will study the associated primes of $I^n$ for $\\forall n > 0$. Moreover, we compute the saturation of $I^n$ for $1 \\leq \\forall n \\leq m$ in the case where $R$ is a Cohen-Macaulay ring and the entries of $M$ are powers of elements that form an sop for $R$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0593","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":"1307.0593","created_at":"2026-05-18T03:19:25.207101+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.0593v1","created_at":"2026-05-18T03:19:25.207101+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0593","created_at":"2026-05-18T03:19:25.207101+00:00"},{"alias_kind":"pith_short_12","alias_value":"JKOSBI5AFGM7","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_16","alias_value":"JKOSBI5AFGM7KZ46","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_8","alias_value":"JKOSBI5A","created_at":"2026-05-18T12:27:49.015174+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/JKOSBI5AFGM7KZ46RYHDBMFVSD","json":"https://pith.science/pith/JKOSBI5AFGM7KZ46RYHDBMFVSD.json","graph_json":"https://pith.science/api/pith-number/JKOSBI5AFGM7KZ46RYHDBMFVSD/graph.json","events_json":"https://pith.science/api/pith-number/JKOSBI5AFGM7KZ46RYHDBMFVSD/events.json","paper":"https://pith.science/paper/JKOSBI5A"},"agent_actions":{"view_html":"https://pith.science/pith/JKOSBI5AFGM7KZ46RYHDBMFVSD","download_json":"https://pith.science/pith/JKOSBI5AFGM7KZ46RYHDBMFVSD.json","view_paper":"https://pith.science/paper/JKOSBI5A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.0593&json=true","fetch_graph":"https://pith.science/api/pith-number/JKOSBI5AFGM7KZ46RYHDBMFVSD/graph.json","fetch_events":"https://pith.science/api/pith-number/JKOSBI5AFGM7KZ46RYHDBMFVSD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JKOSBI5AFGM7KZ46RYHDBMFVSD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JKOSBI5AFGM7KZ46RYHDBMFVSD/action/storage_attestation","attest_author":"https://pith.science/pith/JKOSBI5AFGM7KZ46RYHDBMFVSD/action/author_attestation","sign_citation":"https://pith.science/pith/JKOSBI5AFGM7KZ46RYHDBMFVSD/action/citation_signature","submit_replication":"https://pith.science/pith/JKOSBI5AFGM7KZ46RYHDBMFVSD/action/replication_record"}},"created_at":"2026-05-18T03:19:25.207101+00:00","updated_at":"2026-05-18T03:19:25.207101+00:00"}