{"paper":{"title":"Small ideals in polynomial rings and applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.CO"],"primary_cat":"math.AC","authors_text":"Amichai Lampert","submitted_at":"2023-09-28T20:56:32Z","abstract_excerpt":"Let $\\mathbf{k}$ be a field which is either finite or algebraically closed and let $R = \\mathbf{k}[x_1,\\ldots,x_n].$ We prove that any $g_1,\\ldots,g_s\\in R$ homogeneous of positive degrees $\\le d$ are contained in an ideal generated by an $R_t$-sequence of $\\le A(d)(s+t)^{B(d)}$ homogeneous polynomials of degree $\\le d,$ subject to some restrictions on the characteristic of $\\mathbf{k}.$ This yields effective bounds for new cases of Ananyan and Hochster's theorem A in arXiv:1610.09268 on strength and the codimension of the singular locus. It also implies effective bounds when $d$ equals the ch"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.16847","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2309.16847/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"}