{"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. The first aim of this paper is to study this notion for monomial ideals.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The paper defines the strong persistence index and fluctuation phenomena for colon powers of ideals, then investigates both concepts specifically for monomial ideals.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Monomial ideals possess a finite strong persistence index after which (I^{ℓ+1} : I) equals I^ℓ for all larger ℓ.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3637aadd13fbab1390af835b2570d8ce22bb8b82c8fcaf993de390e71a3b5c88"},"source":{"id":"2604.11475","kind":"arxiv","version":2},"verdict":{"id":"a5988952-016a-4e2f-a07d-a5c125493593","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:39:55.417868Z","strongest_claim":"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. 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"}