2003/07/31 by Thierry Zell
Computer Science · Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Computational Geometry and Mesh Generation #math.AG #math.LO #msc:03C64 #msc:14P10
paper · pdf · doi:10.1007/s00454-004-1112-8
published as Discrete Comput. Geom. 33 (2005) 423--443 · Latex, 23 pages, no figures. v2: Many changes in the exposition and notations in an attempt to be clearer, references added
arxiv created 2004/03/23 · openalex publication_date 2004/10/20 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29
Let A\sub \Rn+r be a set definable in an o-minimal expansion § of the real field, A' \sub \Rr be its projection, and assume that the non-empty fibers Aa \sub \Rn are compact for all a ∈ A' and uniformly bounded, \em i.e. all fibers are contained in a ball of fixed radius B(0,R). If L is the Hausdorff limit of a sequence of fibers Aai, we give an upper-bound for the Betti numbers bk(L) in terms of definable sets explicitly constructed from a fiber Aa. In particular, this allows to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the \em relative closure to construct the o-minimal structure §_\pfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure (X,Y)0 in the special case where Y is empty.