{"paper":{"title":"Dual Polynomials for Collision and Element Distinctness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["quant-ph"],"primary_cat":"cs.CC","authors_text":"Justin Thaler, Mark Bun","submitted_at":"2015-03-25T02:14:41Z","abstract_excerpt":"The approximate degree of a Boolean function $f: \\{-1, 1\\}^n \\to \\{-1, 1\\}$ is the minimum degree of a real polynomial that approximates $f$ to within error $1/3$ in the $\\ell_\\infty$ norm. In an influential result, Aaronson and Shi (J. ACM 2004) proved tight $\\tilde{\\Omega}(n^{1/3})$ and $\\tilde{\\Omega}(n^{2/3})$ lower bounds on the approximate degree of the Collision and Element Distinctness functions, respectively. Their proof was non-constructive, using a sophisticated symmetrization argument and tools from approximation theory.\n  More recently, several open problems in the study of approx"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07261","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"}