{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:LFY6YYVHHTFXAVDDZ5DDL5XLBD","short_pith_number":"pith:LFY6YYVH","schema_version":"1.0","canonical_sha256":"5971ec62a73ccb705463cf4635f6eb08fe8bf834364d8a92c1234b2b1413b7af","source":{"kind":"arxiv","id":"1302.6191","version":3},"attestation_state":"computed","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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1302.6191","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-02-25T18:53:47Z","cross_cats_sorted":[],"title_canon_sha256":"d98f02d09995b9408c0233c1430566e2ebf97f325274f6f10808798c6aaf589b","abstract_canon_sha256":"1383e04e76b4772f674f3dff7cbf1ca0f1c5e004a703059f1f2216c1f8246d99"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:49.932735Z","signature_b64":"AmTJrs8vMGWi0VwI5YImAJwTUO3Yc/bmVLe2iBi7J6x752E99Alvt3cj1xu7wCjzVPu4K+e9yFXhOKL0oJk3Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5971ec62a73ccb705463cf4635f6eb08fe8bf834364d8a92c1234b2b1413b7af","last_reissued_at":"2026-05-18T02:55:49.932044Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:49.932044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1302.6191","created_at":"2026-05-18T02:55:49.932165+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.6191v3","created_at":"2026-05-18T02:55:49.932165+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.6191","created_at":"2026-05-18T02:55:49.932165+00:00"},{"alias_kind":"pith_short_12","alias_value":"LFY6YYVHHTFX","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_16","alias_value":"LFY6YYVHHTFXAVDD","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_8","alias_value":"LFY6YYVH","created_at":"2026-05-18T12:27:51.066281+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LFY6YYVHHTFXAVDDZ5DDL5XLBD","json":"https://pith.science/pith/LFY6YYVHHTFXAVDDZ5DDL5XLBD.json","graph_json":"https://pith.science/api/pith-number/LFY6YYVHHTFXAVDDZ5DDL5XLBD/graph.json","events_json":"https://pith.science/api/pith-number/LFY6YYVHHTFXAVDDZ5DDL5XLBD/events.json","paper":"https://pith.science/paper/LFY6YYVH"},"agent_actions":{"view_html":"https://pith.science/pith/LFY6YYVHHTFXAVDDZ5DDL5XLBD","download_json":"https://pith.science/pith/LFY6YYVHHTFXAVDDZ5DDL5XLBD.json","view_paper":"https://pith.science/paper/LFY6YYVH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.6191&json=true","fetch_graph":"https://pith.science/api/pith-number/LFY6YYVHHTFXAVDDZ5DDL5XLBD/graph.json","fetch_events":"https://pith.science/api/pith-number/LFY6YYVHHTFXAVDDZ5DDL5XLBD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LFY6YYVHHTFXAVDDZ5DDL5XLBD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LFY6YYVHHTFXAVDDZ5DDL5XLBD/action/storage_attestation","attest_author":"https://pith.science/pith/LFY6YYVHHTFXAVDDZ5DDL5XLBD/action/author_attestation","sign_citation":"https://pith.science/pith/LFY6YYVHHTFXAVDDZ5DDL5XLBD/action/citation_signature","submit_replication":"https://pith.science/pith/LFY6YYVHHTFXAVDDZ5DDL5XLBD/action/replication_record"}},"created_at":"2026-05-18T02:55:49.932165+00:00","updated_at":"2026-05-18T02:55:49.932165+00:00"}