{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:KOZDXB3EZMVDDKHDYYWOBNDEPG","short_pith_number":"pith:KOZDXB3E","schema_version":"1.0","canonical_sha256":"53b23b8764cb2a31a8e3c62ce0b464799490c23f526e9cf88eb7a3a986dddd5e","source":{"kind":"arxiv","id":"1503.07705","version":1},"attestation_state":"computed","paper":{"title":"Log-concavity and lower bounds for arithmetic circuits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.AC"],"primary_cat":"cs.CC","authors_text":"Ignacio Garc\\'ia-Marco (LIP), Pascal Koiran (LIP), S\\'ebastien Tavenas","submitted_at":"2015-03-26T12:29:43Z","abstract_excerpt":"One question that we investigate in this paper is, how can we build log-concave polynomials using sparse polynomials as building blocks? More precisely, let $f = \\sum\\_{i = 0}^d a\\_i X^i \\in \\mathbb{R}^+[X]$ be a polynomial satisfying the log-concavity condition $a\\_i^2 \\textgreater{} \\tau a\\_{i-1}a\\_{i+1}$ for every $i \\in \\{1,\\ldots,d-1\\},$ where $\\tau \\textgreater{} 0$. Whenever $f$ can be written under the form $f = \\sum\\_{i = 1}^k \\prod\\_{j = 1}^m f\\_{i,j}$ where the polynomials $f\\_{i,j}$ have at most $t$ monomials, it is clear that $d \\leq k t^m$. Assuming that the $f\\_{i,j}$ have only "},"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":"1503.07705","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-03-26T12:29:43Z","cross_cats_sorted":["cs.DM","math.AC"],"title_canon_sha256":"4dfb5ecc8038edb7cdd7fb11ad9ab5a71e4748f40c0da5baa12b4fb67bf1591a","abstract_canon_sha256":"e9e010202505fbb62c9c4d6c5c4cd887994be41e450a0df92197015df75f7f49"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:53.303352Z","signature_b64":"IKraVD7O+dkp2gGenJyzq0RuMY4uTEMf9swleVdLQTAt18zzqcs/Ou9cp6VZmBFGNSkJO/hELuhhoiQmMJiwDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53b23b8764cb2a31a8e3c62ce0b464799490c23f526e9cf88eb7a3a986dddd5e","last_reissued_at":"2026-05-18T00:52:53.302599Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:53.302599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Log-concavity and lower bounds for arithmetic circuits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.AC"],"primary_cat":"cs.CC","authors_text":"Ignacio Garc\\'ia-Marco (LIP), Pascal Koiran (LIP), S\\'ebastien Tavenas","submitted_at":"2015-03-26T12:29:43Z","abstract_excerpt":"One question that we investigate in this paper is, how can we build log-concave polynomials using sparse polynomials as building blocks? More precisely, let $f = \\sum\\_{i = 0}^d a\\_i X^i \\in \\mathbb{R}^+[X]$ be a polynomial satisfying the log-concavity condition $a\\_i^2 \\textgreater{} \\tau a\\_{i-1}a\\_{i+1}$ for every $i \\in \\{1,\\ldots,d-1\\},$ where $\\tau \\textgreater{} 0$. Whenever $f$ can be written under the form $f = \\sum\\_{i = 1}^k \\prod\\_{j = 1}^m f\\_{i,j}$ where the polynomials $f\\_{i,j}$ have at most $t$ monomials, it is clear that $d \\leq k t^m$. Assuming that the $f\\_{i,j}$ have only "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07705","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":"1503.07705","created_at":"2026-05-18T00:52:53.302696+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.07705v1","created_at":"2026-05-18T00:52:53.302696+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.07705","created_at":"2026-05-18T00:52:53.302696+00:00"},{"alias_kind":"pith_short_12","alias_value":"KOZDXB3EZMVD","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_16","alias_value":"KOZDXB3EZMVDDKHD","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_8","alias_value":"KOZDXB3E","created_at":"2026-05-18T12:29:29.992203+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/KOZDXB3EZMVDDKHDYYWOBNDEPG","json":"https://pith.science/pith/KOZDXB3EZMVDDKHDYYWOBNDEPG.json","graph_json":"https://pith.science/api/pith-number/KOZDXB3EZMVDDKHDYYWOBNDEPG/graph.json","events_json":"https://pith.science/api/pith-number/KOZDXB3EZMVDDKHDYYWOBNDEPG/events.json","paper":"https://pith.science/paper/KOZDXB3E"},"agent_actions":{"view_html":"https://pith.science/pith/KOZDXB3EZMVDDKHDYYWOBNDEPG","download_json":"https://pith.science/pith/KOZDXB3EZMVDDKHDYYWOBNDEPG.json","view_paper":"https://pith.science/paper/KOZDXB3E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.07705&json=true","fetch_graph":"https://pith.science/api/pith-number/KOZDXB3EZMVDDKHDYYWOBNDEPG/graph.json","fetch_events":"https://pith.science/api/pith-number/KOZDXB3EZMVDDKHDYYWOBNDEPG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KOZDXB3EZMVDDKHDYYWOBNDEPG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KOZDXB3EZMVDDKHDYYWOBNDEPG/action/storage_attestation","attest_author":"https://pith.science/pith/KOZDXB3EZMVDDKHDYYWOBNDEPG/action/author_attestation","sign_citation":"https://pith.science/pith/KOZDXB3EZMVDDKHDYYWOBNDEPG/action/citation_signature","submit_replication":"https://pith.science/pith/KOZDXB3EZMVDDKHDYYWOBNDEPG/action/replication_record"}},"created_at":"2026-05-18T00:52:53.302696+00:00","updated_at":"2026-05-18T00:52:53.302696+00:00"}