{"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"}