Communications in Contemporary Mathematics ( IF 1.278 ) Pub Date : 2020-06-22 , DOI: 10.1142/s0219199720500182 Raf Cluckers; Omer Friedland; Yosef Yomdin
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 This is done in a rather general setting, i.e. for the -complement of a polynomial zero-level hypersurface and for the regular level hypersurfaces themselves with no assumptions on the singularities of . The coefficient is the ambient dimension in the first case and in the second case. However, the question of a uniform behavior of the coefficient 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 of dimension away from the -neighborhood of a lower dimensional set , with bound of the form holding uniformly in the complexity of . We also show an analogue for level sets with parameter away from the -neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.