{"paper":{"title":"{\\L}ojasiewicz-type inequalities with explicit exponents for the largest eigenvalue function of real symmetric polynomial matrices","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":"2015-01-07T09:52:43Z","abstract_excerpt":"Let $F(x) := (f_{ij}(x))_{i,j=1,\\ldots,p},$ be a real symmetric polynomial matrix of order $p$ and let $f(x)$ be the largest eigenvalue function of the matrix $F(x).$ We denote by ${\\partial}^\\circ f(x)$ the Clarke subdifferential of $f$ at $x.$ In this paper, we first give the following {\\em nonsmooth} version of \\L ojasiewicz gradient inequality for the function $f$ with an explicit exponent: 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}^\\circ f(x) \\} \\ \\ge \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01419","kind":"arxiv","version":3},"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"}