{"paper":{"title":"Convexifying positive polynomials and sums of squares approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Krzysztof Kurdyka, Stanis{\\l}aw Spodzieja","submitted_at":"2015-07-22T14:05:13Z","abstract_excerpt":"We show that if a polynomial $f\\in \\mathbb{R}[x_1,\\ldots,x_n]$ is nonnegative on a closed basic semialgebraic set $X=\\{x\\in\\mathbb{R}^n:g_1(x)\\ge 0,\\ldots,g_r (x)\\ge 0\\}$, where $g_1,\\ldots,g_r\\in\\mathbb{R}[x_1,\\ldots,x_n]$, then $f$ can be approximated uniformly on compact sets by polynomials of the form $\\sigma_0+\\varphi(g_1) g_1+\\cdots +\\varphi(g_r) g_r$, where $\\sigma_0\\in \\mathbb{R}[x_1,\\ldots,x_n]$ and $\\varphi\\in\\mathbb{R}[t]$ are sums of squares of polynomials. In particular, if $X$ is compact, and $h(x):=R^2-|x|^2 $ is positive on $X$, then $f=\\sigma_{0}+\\sigma_1 h+\\varphi(g_1) g_1+\\c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06191","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"}