{"paper":{"title":"Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities","license":"","headline":"","cross_cats":["math.AT","math.GT"],"primary_cat":"math.AG","authors_text":"Michael Kettner, Saugata Basu","submitted_at":"2007-07-30T15:06:15Z","abstract_excerpt":"We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\\R^\\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in $\\ell$. More precisely, we prove the following. Let $\\R$ be a real closed field and let \\[ {\\mathcal P} = \\{P_1,...,P_m\\} \\subset \\R[Y_1,...,Y_\\ell,X_1,...,X_k], \\] with ${\\rm deg}_Y(P_i) \\leq 2, {\\rm deg}_X(P_i) \\leq d, 1 \\leq i \\leq m$. Let $S \\subset \\R^{\\ell+k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.4333","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"}