{"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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610954","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"}