Topology of definable Hausdorff limits
| dc.creator | Zell, Thierry | |
| dc.date | 2003-07-28 | |
| dc.date | 2004-03-23 | |
| dc.date.accessioned | 2026-07-07T04:59:56Z | |
| dc.date.available | 2026-07-07T04:59:56Z | |
| dc.description | Let $A\sub \R^{n+r}$ be a set definable in an o-minimal expansion $§$ of the real field, $A' \sub \R^r$ be its projection, and assume that the non-empty fibers $A_a \sub \R^n$ are compact for all $a \in A'$ 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 $A_{a_i},$ we give an upper-bound for the Betti numbers $b_k(L)$ in terms of definable sets explicitly constructed from a fiber $A_a.$ 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. | |
| dc.description | Latex, 23 pages, no figures. v2: Many changes in the exposition and notations in an attempt to be clearer, references added | |
| dc.identifier | https://arxiv.org/abs/math/0307369 | |
| dc.identifier | http://arxiv.org/abs/math/0307369 | |
| dc.identifier | Discrete Comput. Geom. 33 (2005) 423--443 | |
| dc.identifier | doi:10.1007/s00454-004-1112-8 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68194 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Logic | |
| dc.subject | 14P10 (Primary); 03C64 (Secondary) | |
| dc.title | Topology of definable Hausdorff limits | |
| dc.type | text |