{"paper":{"title":"A Linear Bound on the Projective Dimension of Height 3 Quadratic Ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Height 3 quadratic ideals have projective dimension bounded linearly by the number of generators.","cross_cats":[],"primary_cat":"math.AC","authors_text":"Jason McCullough, Paolo Mantero, Zachary Greif","submitted_at":"2026-05-15T14:23:20Z","abstract_excerpt":"In 2016, Ananyan and Hochster gave the first proof of a positive answer to Stillman's Question, which asked for a bound on the projective dimension of a graded polynomial ideal purely in terms of the number and degrees of its generators. Explicit formulas for such a bound are limited and often not optimal. In this paper, we give a nearly optimal linear upper bound on the projective dimension of height $3$ ideals generated by any number of degree $2$ homogenous polynomials."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give a nearly optimal linear upper bound on the projective dimension of height 3 ideals generated by any number of degree 2 homogeneous polynomials.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proof relies on the ideal having height exactly 3; if this height condition is relaxed or if the generators are not all quadratic, the linear bound may fail to hold by the same argument.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A linear upper bound on the projective dimension of height 3 quadratic ideals.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Height 3 quadratic ideals have projective dimension bounded linearly by the number of generators.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0aca0c8246bc9ed407cc236fc0d3b10708a85182b97c16a60e603916995d8811"},"source":{"id":"2605.15992","kind":"arxiv","version":1},"verdict":{"id":"f13efc1c-06be-4a88-84e8-6bedf326785a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:26:28.453737Z","strongest_claim":"We give a nearly optimal linear upper bound on the projective dimension of height 3 ideals generated by any number of degree 2 homogeneous polynomials.","one_line_summary":"A linear upper bound on the projective dimension of height 3 quadratic ideals.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The proof relies on the ideal having height exactly 3; if this height condition is relaxed or if the generators are not all quadratic, the linear bound may fail to hold by the same argument.","pith_extraction_headline":"Height 3 quadratic ideals have projective dimension bounded linearly by the number of generators."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15992/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T17:36:25.265703Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:42.462978Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T17:31:18.434073Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:01:55.667085Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"7390048513e3b22cfe3c9c33534e6681443088e1c13511fae6a73bcd3956610c"},"references":{"count":30,"sample":[{"doi":"10.4310/mrl.2012.v19.n1.a18","year":2012,"title":"Ananyan, Tigran and Hochster, Melvin , TITLE =. 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