Doubling coverings via resolution of singularities and preparation

Cluckers, Raf; Friedland, Omer; Yomdin, Yosef

Document type :

Article dans une revue scientifique: Article original

DOI :

10.1142/S0219199720500182

Title :

Doubling coverings via resolution of singularities and preparation

Author(s) :

Cluckers, Raf [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]
Friedland, Omer [Auteur]
Institut de Mathématiques de Jussieu - Paris Rive Gauche [IMJ-PRG (UMR_7586)]
Yomdin, Yosef [Auteur]

Journal title :

Communications in Contemporary Mathematics

Publisher :

World Scientific Publishing

Publication date :

2021-03-01

ISSN :

0219-1997

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

In this paper we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form $$ \kappa({\mathcal ...
Show more >In this paper we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form $$ \kappa({\mathcal U}) \le K_1(\log ({1}/{\delta}))^{K_2} . $$ This is done in a rather general setting, i.e. for the $\delta$-complement of a polynomial zero-level hypersurface $Y_0$ and for the regular level hypersurfaces $Y_c$ themselves with no assumptions on the singularities of $P$. The coefficient $K_2$ is the ambient dimension $n$ in the first case and $n-1$ in the second case. However, the question of a uniform behavior of the coefficient $K_1$ remains open. As a second theme, we confirm in arbitrary dimension the upper bound for the number of a-charts covering a real semi-algebraic set $X$ of dimension $m$ away from the $\delta$-neighborhood of a lower dimensional set $S$, with bound of the form $$ \kappa(\delta) \le C (\log ({1}/{\delta}))^{m} $$ holding uniformly in the complexity of $X$. We also show an analogue for level sets with parameter away from the $\delta$-neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :