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Doubling coverings via resolution of singularities and preparation
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 κ(𝒰)K1log1δK2. This is done in a rather general setting, i.e. for the δ-complement of a polynomial zero-level hypersurface Y0 and for the regular level hypersurfaces Yc themselves with no assumptions on the singularities of P. The coefficient K2 is the ambient dimension n in the first case and n1 in the second case. However, the question of a uniform behavior of the coefficient K1 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 δ-neighborhood of a lower dimensional set S, with bound of the form κ(δ)Clog1δm holding uniformly in the complexity of X. We also show an analogue for level sets with parameter away from the δ-neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.

更新日期:2020-08-09

 

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