{"paper":{"title":"Bounding the radii of balls meeting every connected component of semi-algebraic sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"cs.SC","authors_text":"Marie-Francoise Roy, Saugata Basu","submitted_at":"2009-11-06T19:53:54Z","abstract_excerpt":"We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set $S \\subset \\mathbbm{R}^k$ defined by a quantifier-free formula involving $s$ polynomials in $\\mathbbm{Z}[X_1, ..., X_k]$ having degrees at most $d$, and whose coefficients have bitsizes at most $\\tau$. Our bound is an explicit function of $s, d, k$ and $\\tau$, and does not contain any undetermined constants. We also prove a similar bound on the radius of a ball guaranteed to intersect every connected component of $S$ (including the unbo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.1340","kind":"arxiv","version":1},"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"}