{"paper":{"title":"On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.AG"],"primary_cat":"math.AT","authors_text":"Cordian Riener, Saugata Basu","submitted_at":"2016-10-17T01:42:15Z","abstract_excerpt":"Let $\\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in dimensions $d$ and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $d^{O(d)} s^d k^{\\lfloor d/2 \\rfloor-1} $ on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of $\\mathrm{R}^k$ defined by quantifier-free formulas involving $s$ symmetric polynomials of degrees bounded by $d$, wher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04946","kind":"arxiv","version":5},"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"}