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We say that a positive integer $\\ell_0$ is the strong persistence index of $I$ if $\\ell_0$ is the smallest integer such that $(I^{\\ell+1} :_R I) = I^{\\ell}$ for all $\\ell \\geq \\ell_0$. The first aim of this paper is to study this notion for monomial ideals.\n  We also introduce the notion of fluctuation in colon powers if there exist positive integers $a < b < c$ such that at least one of the following cases occurs:\n  (i) $(I^{a} : I) = I^{a-1}$, $(I^{b} : I) \\neq I^{b-1}$, but $(I^{c} : I) = I^{c-1}$.\n  (ii) $(I^{a} : I) \\neq I^{a-1}$, "},"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":"2604.11475","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-13T13:45:05Z","cross_cats_sorted":[],"title_canon_sha256":"f5a9aa4af8a3bc95c2adbb3a0a0d216f8d8ec4f7a5532c1d4eb610b233bbebfc","abstract_canon_sha256":"66721a8c9ddafaf937dbfd5e52e4ff3fb1740107573fd756140484e95e3c65ae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:01:19.412816Z","signature_b64":"uRMTtRC06BfLvgVVeuOT//F0mvXnuo7Ttl1xtmwz3uccXGdZx5F3FCTWq81kATGy4XH/xjiA1z8GZhUkkxbGBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"217b3fc248bc665af997586532bfb9cc9af169a80459721be186ee9357786eaf","last_reissued_at":"2026-05-25T02:01:19.412041Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:01:19.412041Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strong persistence index and fluctuations in colon powers of monomial ideals","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Monomial ideals possess a finite strong persistence index after which (I^{ℓ+1} : I) equals I^ℓ for all larger ℓ.","cross_cats":[],"primary_cat":"math.AC","authors_text":"Jonathan Toledo, Mehrdad Nasernejad","submitted_at":"2026-04-13T13:45:05Z","abstract_excerpt":"Let $I$ be an ideal in a commutative Noetherian ring $R$. We say that a positive integer $\\ell_0$ is the strong persistence index of $I$ if $\\ell_0$ is the smallest integer such that $(I^{\\ell+1} :_R I) = I^{\\ell}$ for all $\\ell \\geq \\ell_0$. The first aim of this paper is to study this notion for monomial ideals.\n  We also introduce the notion of fluctuation in colon powers if there exist positive integers $a < b < c$ such that at least one of the following cases occurs:\n  (i) $(I^{a} : I) = I^{a-1}$, $(I^{b} : I) \\neq I^{b-1}$, but $(I^{c} : I) = I^{c-1}$.\n  (ii) $(I^{a} : I) \\neq I^{a-1}$, "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Let I be an ideal in a commutative Noetherian ring R. We say that a positive integer ℓ₀ is the strong persistence index of I if ℓ₀ is the smallest integer such that (I^{ℓ+1} :_R I) = I^ℓ for all ℓ ≥ ℓ₀. 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We say that a positive integer ℓ₀ is the strong persistence index of I if ℓ₀ is the smallest integer such that (I^{ℓ+1} :_R I) = I^ℓ for all ℓ ≥ ℓ₀. The first aim of this paper is to study this notion for monomial ideals.","one_line_summary":"The paper defines the strong persistence index and fluctuation phenomena for colon powers of ideals, then investigates both concepts specifically for monomial ideals.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The definitions assume that the strong persistence index exists (i.e., there is a finite smallest ℓ₀ satisfying the eventual equality) and that fluctuations can be meaningfully detected by checking finitely many exponents a < b < c; this is not justified in the abstract and may require the Noetherian hypothesis or specific properties of monomial ideals.","pith_extraction_headline":"Monomial ideals possess a finite strong persistence index after which (I^{ℓ+1} : I) equals I^ℓ for all larger ℓ."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11475/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":"2604.11475","created_at":"2026-05-25T02:01:19.412149+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.11475v2","created_at":"2026-05-25T02:01:19.412149+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.11475","created_at":"2026-05-25T02:01:19.412149+00:00"},{"alias_kind":"pith_short_12","alias_value":"EF5T7QSIXRTF","created_at":"2026-05-25T02:01:19.412149+00:00"},{"alias_kind":"pith_short_16","alias_value":"EF5T7QSIXRTFV6MX","created_at":"2026-05-25T02:01:19.412149+00:00"},{"alias_kind":"pith_short_8","alias_value":"EF5T7QSI","created_at":"2026-05-25T02:01:19.412149+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/EF5T7QSIXRTFV6MXLBSTFP5ZZS","json":"https://pith.science/pith/EF5T7QSIXRTFV6MXLBSTFP5ZZS.json","graph_json":"https://pith.science/api/pith-number/EF5T7QSIXRTFV6MXLBSTFP5ZZS/graph.json","events_json":"https://pith.science/api/pith-number/EF5T7QSIXRTFV6MXLBSTFP5ZZS/events.json","paper":"https://pith.science/paper/EF5T7QSI"},"agent_actions":{"view_html":"https://pith.science/pith/EF5T7QSIXRTFV6MXLBSTFP5ZZS","download_json":"https://pith.science/pith/EF5T7QSIXRTFV6MXLBSTFP5ZZS.json","view_paper":"https://pith.science/paper/EF5T7QSI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.11475&json=true","fetch_graph":"https://pith.science/api/pith-number/EF5T7QSIXRTFV6MXLBSTFP5ZZS/graph.json","fetch_events":"https://pith.science/api/pith-number/EF5T7QSIXRTFV6MXLBSTFP5ZZS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EF5T7QSIXRTFV6MXLBSTFP5ZZS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EF5T7QSIXRTFV6MXLBSTFP5ZZS/action/storage_attestation","attest_author":"https://pith.science/pith/EF5T7QSIXRTFV6MXLBSTFP5ZZS/action/author_attestation","sign_citation":"https://pith.science/pith/EF5T7QSIXRTFV6MXLBSTFP5ZZS/action/citation_signature","submit_replication":"https://pith.science/pith/EF5T7QSIXRTFV6MXLBSTFP5ZZS/action/replication_record"}},"created_at":"2026-05-25T02:01:19.412149+00:00","updated_at":"2026-05-25T02:01:19.412149+00:00"}