{"paper":{"title":"Relating F-Signature and F-Splitting Ratio of Pairs Using Left-Derivatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Eric Canton","submitted_at":"2012-06-19T19:17:30Z","abstract_excerpt":"We first relate an approximate $n^{th}$-order left derivative of $s(R, f^t)$ at the F-pure threshold $c$ to the F-splitting ratio $r_F(R, f^c)$. Next, we apply the methods developed by Monsky and Teixeira in their investigation of syzygy gaps and $p$-fractals to obtain uniform convergence of the F-signature when $f$ is a product of distinct linear polynomials in two variables. Finally, we explicitly compute the F-signature function for several examples using Macaulay2 code outlined in the last section of this paper."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4293","kind":"arxiv","version":4},"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"}