{"paper":{"title":"Approximating real-rooted and stable polynomials, with combinatorial applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.CA"],"primary_cat":"math.CO","authors_text":"Alexander Barvinok","submitted_at":"2018-06-19T18:01:23Z","abstract_excerpt":"Let $p(x)=a_0 + a_1 x + \\ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \\leq -\\delta$ for some $0<\\delta <1$. We show that for any $0 < \\epsilon <1$, the value of $p(1)$ is determined within relative error $\\epsilon$ by the coefficients $a_k$ with $k \\leq {c \\over \\sqrt{\\delta}} \\ln {n \\over \\epsilon \\sqrt{ \\delta}}$ for some absolute constant $c > 0$. Consequently, if $m_k(G)$ is the number of matchings with $k$ edges in a graph $G$, then for any $0 < \\epsilon < 1$, the total number $M(G)=m_0(G)+m_1(G) + \\ldots $ of matchings is determined within relative error $\\epsil"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07404","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"}