{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:523XQXQ37K5AATLZ64GZZD3DIX","short_pith_number":"pith:523XQXQ3","schema_version":"1.0","canonical_sha256":"eeb7785e1bfaba004d79f70d9c8f6345d0448f95c3dc56b7036a85d3c467cdf0","source":{"kind":"arxiv","id":"math/0610954","version":2},"attestation_state":"computed","paper":{"title":"A sharper estimate on the Betti numbers of sets defined by quadratic inequalities","license":"","headline":"","cross_cats":["math.AT"],"primary_cat":"math.AG","authors_text":"Michael Kettner, Saugata Basu","submitted_at":"2006-10-31T03:46:02Z","abstract_excerpt":"In this paper we consider the problem of bounding the Betti numbers, $b_i(S)$, of a semi-algebraic set $S \\subset \\R^k$ defined by polynomial inequalities $P_1 \\geq 0,...,P_s \\geq 0$, where $P_i \\in \\R[X_1,...,X_k]$ and $\\deg(P_i) \\leq 2$, for $1 \\leq i \\leq s$. We prove that for $0\\le i\\le k-1$, \\[ b_i(S) \\le{1/2}(\\sum_{j=0}^{min\\{s,k-i\\}}{{s}\\choose j}{{k+1}\\choose {j}}2^{j}). \\] In particular, for $2\\le s\\le \\frac{k}{2}$, we have \\[ b_i(S)\\le {1/2} 3^{s}{{k+1}\\choose {s}} \\leq {1/2} (\\frac{3e(k+1)}{s})^s. \\] This improves the bound of $k^{O(s)}$ proved by Barvinok. This improvement is made "},"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":"math/0610954","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AG","submitted_at":"2006-10-31T03:46:02Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"8e81f697a3a5ec73d3a871d58dc09d50835e45b06c1fcfa575ea92bc3166a2d4","abstract_canon_sha256":"ae7faeeb206e16ce3435508439a4dc3ba0d38e1b55d74923cea8181c9eb4a87c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:20.593495Z","signature_b64":"wv2xbfTaMwK4lPSfRoxBeBQx8xthvPeGF0UNDApjmQXXH/B31VKNTsquxNmBvrpUOXtbkI0sZmyHl7v/c/l0Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eeb7785e1bfaba004d79f70d9c8f6345d0448f95c3dc56b7036a85d3c467cdf0","last_reissued_at":"2026-05-18T04:28:20.593057Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:20.593057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A sharper estimate on the Betti numbers of sets defined by quadratic inequalities","license":"","headline":"","cross_cats":["math.AT"],"primary_cat":"math.AG","authors_text":"Michael Kettner, Saugata Basu","submitted_at":"2006-10-31T03:46:02Z","abstract_excerpt":"In this paper we consider the problem of bounding the Betti numbers, $b_i(S)$, of a semi-algebraic set $S \\subset \\R^k$ defined by polynomial inequalities $P_1 \\geq 0,...,P_s \\geq 0$, where $P_i \\in \\R[X_1,...,X_k]$ and $\\deg(P_i) \\leq 2$, for $1 \\leq i \\leq s$. We prove that for $0\\le i\\le k-1$, \\[ b_i(S) \\le{1/2}(\\sum_{j=0}^{min\\{s,k-i\\}}{{s}\\choose j}{{k+1}\\choose {j}}2^{j}). \\] In particular, for $2\\le s\\le \\frac{k}{2}$, we have \\[ b_i(S)\\le {1/2} 3^{s}{{k+1}\\choose {s}} \\leq {1/2} (\\frac{3e(k+1)}{s})^s. \\] This improves the bound of $k^{O(s)}$ proved by Barvinok. 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