{"paper":{"title":"Refined bounds on the number of connected components of sign conditions on a variety","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Sal Barone, Saugata Basu","submitted_at":"2011-04-04T17:38:28Z","abstract_excerpt":"Let $\\R$ be a real closed field, $\\mathcal{P},\\mathcal{Q} \\subset \\R[X_1,...,X_k]$ finite subsets of polynomials, with the degrees of the polynomials in $\\mathcal{P}$ (resp. $\\mathcal{Q}$) bounded by $d$ (resp. $d_0$). Let $V \\subset \\R^k$ be the real algebraic variety defined by the polynomials in $\\mathcal{Q}$ and suppose that the real dimension of $V$ is bounded by $k'$. We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family $\\mathcal{P}$ on $V$ is bounded by $$ \\displaylines{\\sum_{j=0}^{k'}4^j{s +1\\choose j}F_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.0636","kind":"arxiv","version":4},"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"}