{"paper":{"title":"Milnor and Tjurina numbers for a hypersurface germ with isolated singularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Yongqiang Liu","submitted_at":"2017-08-31T14:01:36Z","abstract_excerpt":"Assume that $f:(\\mathbb{C}^n,0) \\to (\\mathbb{C},0)$ is an analytic function germ at the origin with only isolated singularity. Let $\\mu$ and $\\tau$ be the corresponding Milnor and Tjurina numbers. We show that $\\dfrac{\\mu}{\\tau} \\leq n$. As an application, we give a lower bound for the Tjurina number in terms of $n$ and the multiplicity of $f$ at the origin."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.09716","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"}