{"paper":{"title":"Vanishing-Error Approximate Degree and QMA Complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Alexander A. Sherstov, Justin Thaler","submitted_at":"2019-09-16T21:59:01Z","abstract_excerpt":"The $\\epsilon$-approximate degree of a function $f\\colon X \\to \\{0, 1\\}$ is the least degree of a multivariate real polynomial $p$ such that $|p(x)-f(x)| \\leq \\epsilon$ for all $x \\in X$. We determine the $\\epsilon$-approximate degree of the element distinctness function, the surjectivity function, and the permutation testing problem, showing they are $\\Theta(n^{2/3} \\log^{1/3}(1/\\epsilon))$, $\\tilde\\Theta(n^{3/4} \\log^{1/4}(1/\\epsilon))$, and $\\Theta(n^{1/3} \\log^{2/3}(1/\\epsilon))$, respectively. Previously, these bounds were known only for constant $\\epsilon.$\n  We also derive a connection "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.07498","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1909.07498/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}