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The integration is in the unit cube $[0,1]^d$ for a multivariate polynomial, which has format $f(x_1,\\cdots, x_d)=p_1(x_1,\\cdots, x_d)p_2(x_1,\\cdots, x_d)\\cdots p_k(x_1,\\cdots, x_d)$, where each $p_i(x_1,\\cdots, x_d)=\\sum_{j=1}^d q_j(x_j)$ with all single variable polynomials $q_j(x_j)$ of degree at most two and constant coefficients. We show that there is no any factor polynomial time approximation for the integration $\\int_{[0,1]^d}f(x_1,\\cdots,x_d)d_{x_1}\\cdots d_{x_d}"},"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":"1012.2377","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2010-12-10T20:51:16Z","cross_cats_sorted":[],"title_canon_sha256":"8e2ad3b366a2ff607749551a3454394dd3e5497b9739ddd7708019f38519b632","abstract_canon_sha256":"c32a6b3f11368879cb5a5b83d3d44663046a7a971c935d4a1cbb1db7d4fa102f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:33:38.000817Z","signature_b64":"1R1S7FXJRU4sz0OOzVpn7HGJbqb5JOIuLWX7c3TjFRD027atEqDdA+e7zUCBlGSSZNyaVG+BgDUZitt+BhpaBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"007d7c8793cb27a315f383ff408619a74408ec7c84b00b2ae2626dd703b5dd6b","last_reissued_at":"2026-05-18T04:33:38.000253Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:33:38.000253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multivariate Polynomial Integration and Derivative Are Polynomial Time Inapproximable unless P=NP","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Bin Fu","submitted_at":"2010-12-10T20:51:16Z","abstract_excerpt":"We investigate the complexity of integration and derivative for multivariate polynomials in the standard computation model. The integration is in the unit cube $[0,1]^d$ for a multivariate polynomial, which has format $f(x_1,\\cdots, x_d)=p_1(x_1,\\cdots, x_d)p_2(x_1,\\cdots, x_d)\\cdots p_k(x_1,\\cdots, x_d)$, where each $p_i(x_1,\\cdots, x_d)=\\sum_{j=1}^d q_j(x_j)$ with all single variable polynomials $q_j(x_j)$ of degree at most two and constant coefficients. We show that there is no any factor polynomial time approximation for the integration $\\int_{[0,1]^d}f(x_1,\\cdots,x_d)d_{x_1}\\cdots d_{x_d}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2377","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1012.2377","created_at":"2026-05-18T04:33:38.000337+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.2377v1","created_at":"2026-05-18T04:33:38.000337+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2377","created_at":"2026-05-18T04:33:38.000337+00:00"},{"alias_kind":"pith_short_12","alias_value":"AB6XZB4TZMT2","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_16","alias_value":"AB6XZB4TZMT2GFPT","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_8","alias_value":"AB6XZB4T","created_at":"2026-05-18T12:26:05.355336+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/AB6XZB4TZMT2GFPTQP7UBBQZU5","json":"https://pith.science/pith/AB6XZB4TZMT2GFPTQP7UBBQZU5.json","graph_json":"https://pith.science/api/pith-number/AB6XZB4TZMT2GFPTQP7UBBQZU5/graph.json","events_json":"https://pith.science/api/pith-number/AB6XZB4TZMT2GFPTQP7UBBQZU5/events.json","paper":"https://pith.science/paper/AB6XZB4T"},"agent_actions":{"view_html":"https://pith.science/pith/AB6XZB4TZMT2GFPTQP7UBBQZU5","download_json":"https://pith.science/pith/AB6XZB4TZMT2GFPTQP7UBBQZU5.json","view_paper":"https://pith.science/paper/AB6XZB4T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.2377&json=true","fetch_graph":"https://pith.science/api/pith-number/AB6XZB4TZMT2GFPTQP7UBBQZU5/graph.json","fetch_events":"https://pith.science/api/pith-number/AB6XZB4TZMT2GFPTQP7UBBQZU5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AB6XZB4TZMT2GFPTQP7UBBQZU5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AB6XZB4TZMT2GFPTQP7UBBQZU5/action/storage_attestation","attest_author":"https://pith.science/pith/AB6XZB4TZMT2GFPTQP7UBBQZU5/action/author_attestation","sign_citation":"https://pith.science/pith/AB6XZB4TZMT2GFPTQP7UBBQZU5/action/citation_signature","submit_replication":"https://pith.science/pith/AB6XZB4TZMT2GFPTQP7UBBQZU5/action/replication_record"}},"created_at":"2026-05-18T04:33:38.000337+00:00","updated_at":"2026-05-18T04:33:38.000337+00:00"}