{"paper":{"title":"Tangencies and Polynomial Optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Tien-Son Pham","submitted_at":"2019-02-16T04:45:06Z","abstract_excerpt":"Given a polynomial function $f \\colon \\mathbb{R}^n \\rightarrow \\mathbb{R}$ and a unbounded basic closed semi-algebraic set $S \\subset \\mathbb{R}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms of the so-called {\\em tangency variety} of $f$ on $S$:\n  (i) The $f$ is bounded from below on $S;$\n  (ii) The $f$ attains its infimum on $S;$\n  (iii) The sublevel set $\\{x \\in S \\ | \\ f(x) \\le \\lambda\\}$ for $\\lambda \\in \\mathbb{R}$ is compact;\n  (iv) The $f$ is coercive on $S.$\n  Besides, we also provide some stability criteria for boundedness and coercivity"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.06041","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"}