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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1812.04883","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-12-12T10:27:27Z","cross_cats_sorted":[],"title_canon_sha256":"4e1544d5c750055e863745b49e4650d3e7b78a5e8053c984c5c0dde5bc8b9ee0","abstract_canon_sha256":"581b20b10d6a8243bed00f6a56607ebbc3dc248e2768b73af6776f71c94eed2d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:27.073600Z","signature_b64":"n9+WM13TYfazyyqPX8ytCRKKlPFtUfPp9j5VTZxyb+ftn0xXQOKE3rgKVql1dYvYw+V/CWtx47H/9U/ID+YaCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fcc224e76f95e8e89f145312462a41b933ffcf32ab2f7d4ade07eb6678f5512","last_reissued_at":"2026-05-17T23:58:27.072901Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:27.072901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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