{"paper":{"title":"On the {\\L}ojasiewicz exponent, special direction and maximal polar quotient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Andrzej Lenarcik","submitted_at":"2011-12-23T12:55:35Z","abstract_excerpt":"For a local singular plane curve germ $f(X,Y)=0$ we characterize all nonsingular $\\lambda\\in\\bbC\\{X,Y\\}$ such that the {\\L}ojasiewicz exponent of $\\grad\\,f$ is not attained on the polar curve $\\bJ(\\lambda,f)=0$. When $f$ is not Morse we prove that for the same $\\lambda$'s the maximal polar quotient $q_0(f,\\lambda)$ is strictly less than its generic value $q_0(f)$. Our main tool is the Eggers tree of singularity constructed as a decorated graph of relations between balls in the space of branches defined by using a logarithmic distance."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5578","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":""},"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"}