{"paper":{"title":"On the number of topological types occurring in a parametrized family of arrangements","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.CO","authors_text":"Saugata Basu","submitted_at":"2007-04-03T05:51:58Z","abstract_excerpt":"Let ${\\mathcal S}(\\R)$ be an o-minimal structure over $\\R$, $T \\subset \\R^{k_1+k_2+\\ell}$ a closed definable set, and $$ \\displaylines{\\pi_1: \\R^{k_1+k_2+\\ell}\\to \\R^{k_1 + k_2}, \\pi_2: \\R^{k_1+k_2+\\ell}\\to \\R^{\\ell}, \\ \\pi_3: \\R^{k_1 + k_2} \\to \\R^{k_2}} $$ the projection maps.\n  For any collection ${\\mathcal A} = \\{A_1,...,A_n\\}$ of subsets of $\\R^{k_1+k_2}$, and $\\z \\in \\R^{k_2}$, let $\\A_\\z$ denote the collection of subsets of $\\R^{k_1}$, $\\{A_{1,\\z},..., A_{n,\\z}\\}$, where $A_{i,\\z} = A_i \\cap \\pi_3^{-1}(\\z), 1 \\leq i \\leq n$. We prove that there exists a constant $C = C(T) > 0,$ such tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0704.0295","kind":"arxiv","version":3},"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"}