{"paper":{"title":"Real root finding for determinants of linear matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"cs.SC","authors_text":"Didier Henrion, Mohab Safey El Din, Simone Naldi","submitted_at":"2014-12-18T14:28:36Z","abstract_excerpt":"Let $\\A_0, \\A_1, \\ldots, \\A_n$ be given square matrices of size $m$ with rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal variety $\\{\\X \\in\\RR^n \\: :\\: \\det(\\A_0+x_1\\A_1+\\cdots+x_n\\A_n)=0\\}$. Such a problem finds applications in many areas such as control theory, computational geometry, optimization, etc. Using standard complexity results this problem can be solved using $m^{O(n)}$ arithmetic operations. Under some genericity assumptions on the coefficients of the matrices, we provide an algorithm solving this "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.5873","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"}