{"paper":{"title":"On the number of homotopy types of fibres of a definable map","license":"","headline":"","cross_cats":["math.LO"],"primary_cat":"math.AG","authors_text":"Nicolai Vorobjov, Saugata Basu","submitted_at":"2006-05-18T16:49:04Z","abstract_excerpt":"In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map, in terms of the format of its graph. In particular we show that if a semi-algebraic set $S \\subset {\\R}^{m+n}$, where $\\R$ is a real closed field, is defined by a Boolean formula with $s$ polynomials of degrees less than $d$, and $\\pi: {\\R}^{m+n} \\to {\\R}^n$ is the projection on a subspace, then the number of different homotopy types of fibres of $\\pi$ does not exceed $s^{2(m+1)n}(2^m nd)^{O(nm)}$. As applications of our main results we prove single exponential boun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0605517","kind":"arxiv","version":2},"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"}