{"paper":{"title":"Multiplicity bounds in graded rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Craig Huneke, Kei-ichi Watanabe, Shunsuke Takagi","submitted_at":"2009-12-21T15:06:57Z","abstract_excerpt":"The $F$-threshold $c^J(\\a)$ of an ideal $\\a$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\\a$ with the Frobenius powers of $J$. We study a conjecture formulated in an earlier paper \\cite{HMTW} by the same authors together with M. Musta\\c{t}\\u{a}, which bounds $c^J(\\a)$ in terms of the multiplicities $e(\\a)$ and $e(J)$, when $\\a$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\\a$ and $J$ are generated by homogeneous systems of parameters in a Noetherian graded $k$-algeb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.3853","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}