{"paper":{"title":"Dual Lower Bounds for Approximate Degree and Markov-Bernstein Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Justin Thaler, Mark Bun","submitted_at":"2013-02-25T18:53:47Z","abstract_excerpt":"The $\\epsilon$-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 $\\epsilon$ in the $\\ell_\\infty$ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the $\\epsilon$-approximate degree of the two-level AND-OR tree for any constant $\\epsilon > 0$. We show that this quantity is $\\Theta(\\sqrt{n})$, closing a line of incrementally larger lower bounds. The same lower bound was "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.6191","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"}