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Pasechnik, Marie-Francoise Roy, Saugata Basu","submitted_at":"2007-08-27T01:31:17Z","abstract_excerpt":"Let $\\R$ be a real closed field, $ {\\mathcal Q} \\subset \\R[Y_1,...,Y_\\ell,X_1,...,X_k], $ with $ \\deg_{Y}(Q) \\leq 2, \\deg_{X}(Q) \\leq d, Q \\in {\\mathcal Q}, #({\\mathcal Q})=m,$ and $ {\\mathcal P} \\subset \\R[X_1,...,X_k] $ with $\\deg_{X}(P) \\leq d, P \\in {\\mathcal P}, #({\\mathcal P})=s$, and $S \\subset \\R^{\\ell+k}$ a semi-algebraic set defined by a Boolean formula without negations, with atoms $P=0, P \\geq 0, P \\leq 0, P \\in {\\mathcal P} \\cup {\\mathcal Q}$. We prove that the sum of the Betti numbers of $S$ is bounded by \\[ \\ell^2 (O(s+\\ell+m)\\ell d)^{k+2m}. \\] This is a common generalization of"},"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":"0708.3522","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2007-08-27T01:31:17Z","cross_cats_sorted":["cs.SC","math.AT","math.GT"],"title_canon_sha256":"34945a7cb5a32c8f8dab1914931e0466765ba4ffd06f2b856fab98fca0866a14","abstract_canon_sha256":"a0bc2d9f3efdef242d53439ad0fd375964aa7e165fab1a1333ad91cfc752dbfb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:06.945985Z","signature_b64":"cnzTWa+9kBx0r2uA5HjtMGe/WwUxB3Q0nP1dPQ7Ab90Oj5LaA08WcQvmlpvz1siIMuU9yoOfzfAeF43WEsWTBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2942355c97afa890c6ee593ed801eeeb93def401b98848602916fe80816c025","last_reissued_at":"2026-05-18T04:39:06.945278Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:06.945278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounding the Betti numbers and computing the Euler-Poincar\\'e characteristic of semi-algebraic sets defined by partly quadratic systems of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SC","math.AT","math.GT"],"primary_cat":"math.AG","authors_text":"Dmitrii V. Pasechnik, Marie-Francoise Roy, Saugata Basu","submitted_at":"2007-08-27T01:31:17Z","abstract_excerpt":"Let $\\R$ be a real closed field, $ {\\mathcal Q} \\subset \\R[Y_1,...,Y_\\ell,X_1,...,X_k], $ with $ \\deg_{Y}(Q) \\leq 2, \\deg_{X}(Q) \\leq d, Q \\in {\\mathcal Q}, #({\\mathcal Q})=m,$ and $ {\\mathcal P} \\subset \\R[X_1,...,X_k] $ with $\\deg_{X}(P) \\leq d, P \\in {\\mathcal P}, #({\\mathcal P})=s$, and $S \\subset \\R^{\\ell+k}$ a semi-algebraic set defined by a Boolean formula without negations, with atoms $P=0, P \\geq 0, P \\leq 0, P \\in {\\mathcal P} \\cup {\\mathcal Q}$. We prove that the sum of the Betti numbers of $S$ is bounded by \\[ \\ell^2 (O(s+\\ell+m)\\ell d)^{k+2m}. \\] This is a common generalization of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.3522","kind":"arxiv","version":2},"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":"0708.3522","created_at":"2026-05-18T04:39:06.945394+00:00"},{"alias_kind":"arxiv_version","alias_value":"0708.3522v2","created_at":"2026-05-18T04:39:06.945394+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0708.3522","created_at":"2026-05-18T04:39:06.945394+00:00"},{"alias_kind":"pith_short_12","alias_value":"2KKCGVOJPL5I","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"2KKCGVOJPL5ISDDO","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"2KKCGVOJ","created_at":"2026-05-18T12:25:54.717736+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/2KKCGVOJPL5ISDDO4WJ63AA652","json":"https://pith.science/pith/2KKCGVOJPL5ISDDO4WJ63AA652.json","graph_json":"https://pith.science/api/pith-number/2KKCGVOJPL5ISDDO4WJ63AA652/graph.json","events_json":"https://pith.science/api/pith-number/2KKCGVOJPL5ISDDO4WJ63AA652/events.json","paper":"https://pith.science/paper/2KKCGVOJ"},"agent_actions":{"view_html":"https://pith.science/pith/2KKCGVOJPL5ISDDO4WJ63AA652","download_json":"https://pith.science/pith/2KKCGVOJPL5ISDDO4WJ63AA652.json","view_paper":"https://pith.science/paper/2KKCGVOJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0708.3522&json=true","fetch_graph":"https://pith.science/api/pith-number/2KKCGVOJPL5ISDDO4WJ63AA652/graph.json","fetch_events":"https://pith.science/api/pith-number/2KKCGVOJPL5ISDDO4WJ63AA652/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2KKCGVOJPL5ISDDO4WJ63AA652/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2KKCGVOJPL5ISDDO4WJ63AA652/action/storage_attestation","attest_author":"https://pith.science/pith/2KKCGVOJPL5ISDDO4WJ63AA652/action/author_attestation","sign_citation":"https://pith.science/pith/2KKCGVOJPL5ISDDO4WJ63AA652/action/citation_signature","submit_replication":"https://pith.science/pith/2KKCGVOJPL5ISDDO4WJ63AA652/action/replication_record"}},"created_at":"2026-05-18T04:39:06.945394+00:00","updated_at":"2026-05-18T04:39:06.945394+00:00"}