{"paper":{"title":"{\\L}ojasiewicz inequalities with explicit exponent for smallest singular value functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Si Tiep Dinh, Tien Son Pham","submitted_at":"2016-04-11T06:21:50Z","abstract_excerpt":"Let $F(x) := (f_{ij}(x))_{i=1,\\ldots,p; j=1,\\ldots,q},$ be a ($p\\times q$)-real polynomial matrix and let $f(x)$ be the smallest singular value function of $F(x).$ In this paper, we first give the following {\\em nonsmooth} version of \\L ojasiewicz gradient inequality for the function $f$ with an explicit exponent: {\\em For any $\\bar x\\in \\Bbb R^n$, there exist $c > 0$ and $\\epsilon > 0$ such that we have for all $\\|x - \\bar{x}\\| < \\epsilon,$ \\begin{equation*} \\inf \\{ \\| w \\| \\ : \\ w \\in {\\partial} f(x) \\} \\ \\ge \\ c\\, |f(x)-f(\\bar x)|^{1 - \\frac{2}{\\mathscr R(n+p,2d+2)}}, \\end{equation*} where "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02805","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"}