{"paper":{"title":"Numerical characterizations for integral dependence of graded modules","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Adic and density functions give criteria for integral dependence of graded torsion-free modules.","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Sudeshna Roy, Suprajo Das, Vijaylaxmi Trivedi","submitted_at":"2026-05-13T23:40:28Z","abstract_excerpt":"In this paper we construct {\\em adic}, {\\em saturated} and $\\varepsilon$-density functions for a torsion-free module in a graded setup. Then we give some simple criteria for checking the integral dependence of two graded modules $N\\subseteq M$ in terms of various well-studied invariants."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct adic, saturated and ε-density functions for a torsion-free module in a graded setup. Then we give some simple criteria for checking the integral dependence of two graded modules N⊆M in terms of various well-studied invariants.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The constructions and criteria apply specifically to torsion-free modules in a graded setup; the abstract provides no indication of how the functions behave or whether the criteria extend when torsion is present or the grading is absent.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"New density functions and simple criteria are given to characterize integral dependence of graded modules via well-studied invariants.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Adic and density functions give criteria for integral dependence of graded torsion-free modules.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d681c1443bb61d4038154d7560482f2c4ad5d04fc1f5a67dd3c8c5709d768acb"},"source":{"id":"2605.14203","kind":"arxiv","version":1},"verdict":{"id":"1e741437-aaf2-4eea-84ee-dd225b2b0a79","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:31:22.487178Z","strongest_claim":"We construct adic, saturated and ε-density functions for a torsion-free module in a graded setup. Then we give some simple criteria for checking the integral dependence of two graded modules N⊆M in terms of various well-studied invariants.","one_line_summary":"New density functions and simple criteria are given to characterize integral dependence of graded modules via well-studied invariants.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The constructions and criteria apply specifically to torsion-free modules in a graded setup; the abstract provides no indication of how the functions behave or whether the criteria extend when torsion is present or the grading is absent.","pith_extraction_headline":"Adic and density functions give criteria for integral dependence of graded torsion-free modules."},"references":{"count":20,"sample":[{"doi":"","year":2024,"title":"Y. Cid-Ruiz. Polar multiplicities and integral dependence.International Mathematics Research Notices, 2024(17):12201–12218, 2024. 1","work_id":"3e245fb6-64be-4e56-841c-92412aa0676f","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Y. Cid-Ruiz, C. Polini, and B. Ulrich. Multidegrees, families, and integral dependence.arXiv preprint arXiv:2405.07000, 2024. 1","work_id":"2b3ebc04-e107-4667-8308-74805d450805","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"Cutkosky,Asymptotic growth of saturated powers and epsilon multiplicityMath","work_id":"4366598a-0c60-40d3-b0f1-512ff94e1d68","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"Cutkosky,Asymptotic multiplicities of graded families of ideals and linear series, Advances in Mathematics, 264 (2014),55-113, Elsevier","work_id":"284900f9-0a64-487c-988b-88c9b3c63f3e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"S. Das, S. Roy, and V. Trivedi. Density functions for epsilon multiplicity and families of ideals.Journal of the London Mathematical Society. Second Series, 111(4):Paper No. e70155, 51, 2025. 1, 3, 5,","work_id":"a83b245f-1122-460f-b80c-e04fd3b9e398","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":20,"snapshot_sha256":"9723429545797e32531dce765ddb9161633a887dff098b18e9889b7685b2e29e","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"}