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Doubling coverings via resolution of singularities and preparation
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2020-06-22 , DOI: 10.1142/s0219199720500182
Raf Cluckers 1, 2 , Omer Friedland 3 , Yosef Yomdin 4
Affiliation  

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 [Formula: see text] This is done in a rather general setting, i.e. for the [Formula: see text]-complement of a polynomial zero-level hypersurface [Formula: see text] and for the regular level hypersurfaces [Formula: see text] themselves with no assumptions on the singularities of [Formula: see text]. The coefficient [Formula: see text] is the ambient dimension [Formula: see text] in the first case and [Formula: see text] in the second case. However, the question of a uniform behavior of the coefficient [Formula: see text] 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 [Formula: see text] of dimension [Formula: see text] away from the [Formula: see text]-neighborhood of a lower dimensional set [Formula: see text], with bound of the form [Formula: see text] holding uniformly in the complexity of [Formula: see text]. We also show an analogue for level sets with parameter away from the [Formula: see text]-neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.

中文翻译:

通过奇点的解决和准备加倍覆盖

在本文中,我们提供了两种(密切相关)情况下复杂度的渐近上限。我们确认总的加倍覆盖,而不仅仅是链的预期边界 [公式:参见文本] 这是在相当一般的设置中完成的,即对于 [公式:参见文本]-多项式零的补码-水平超曲面[公式:见文本]和常规水平超曲面[公式:见文本]本身,对[公式:见文本]的奇点没有任何假设。系数[公式:见文]是第一种情况下的环境维度[公式:见文]和第二种情况下的[公式:见文]。然而,系数 [公式:见正文] 的统一行为的问题仍然悬而未决。作为第二个主题,我们在任意维度上确认覆盖一个真正的半代数集 [公式:见文本] 的 a-charts 数量的上限,其维度 [公式:见文本] 远离 [公式:见文本] - 邻域维集[公式:见文本],在[公式:见文本]的复杂性中统一保持形式[公式:见文本]的界限。我们还展示了水平集的类似物,其参数远离低维集的 [公式:见文本]-邻域。更一般地,对于实子解析集和实次幂子解析集也可以获得边界。见正文]。我们还展示了水平集的类似物,其参数远离低维集的 [公式:见文本]-邻域。更一般地,对于实子解析集和实次幂子解析集也可以获得边界。见正文]。我们还展示了水平集的类似物,其参数远离低维集的 [公式:见文本]-邻域。更一般地,对于实子解析集和实次幂子解析集也可以获得边界。
更新日期:2020-06-22
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