{"paper":{"title":"Effective {\\L}ojasiewicz gradient inequality and finite determinacy of non-isolated Nash function singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Beata Osi\\'nska-Ulrych, Grzegorz Skalski, Stanis{\\l}aw Spodzieja","submitted_at":"2018-12-12T10:27:27Z","abstract_excerpt":"Let $X\\subset \\mathbb{R}^n$ be a compact semialgebraic set and let $f:X\\to \\mathbb{R}$ be a nonzero Nash function. We give a Solern\\'o and D'Acunto-Kurdyka type estimation of the exponent $\\varrho\\in[0,1)$ in the {\\L}ojasiewicz gradient inequality $|\\nabla f(x)|\\ge C|f(x)|^\\varrho$ for $x\\in X$, $|f(x)|<\\varepsilon$ for some constants $C,\\varepsilon>0$, in terms of the degree of a polynomial $P$ such that $P(x,f(x))=0$, $x\\in X$. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash functions singularities"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04883","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"}