奇异隐式与反函数定理。具有正常平坦中心的强分辨率
Singular implicit and inverse function theorems. Strong resolution with normally flat centers
摘要：
基于Hironaka，Bierstone Milman，Malgrange等人的思想，我们将逆函数和隐函数定理（在微分，解析和代数环境中）推广到具有较大多重性（或理想）的函数集。这允许一个描述的奇点由一个有限集的发电机或理想在一个更简单的形式。在特殊的Cohen-Macaulay情形下，我们得到了反函数定理的一个奇异模拟。奇异隐函数定理与Weierstrass-Hironaka-Malgrange除法和准备定理的一个扩展版本密切相关。本文的主要动机来自去奇异化问题。作为一个例子所使用的技术，我们给我们的定理去奇异扩展Hironaka正常平坦，塞缪尔分层和希尔伯特-塞缪尔函数的一些结果的一些应用。本文中引入的Samuel层的标准基的概念（受Bierstone-Milman和Hironaka构造的启发）使我们能够以相对简单的方式描述Samuel层的奇点。它导致了一个典型的减少与正常平坦中心的强Hironaka desingularization到一个所谓的决议的标志理想。此外，在特征零点处，沿着Samuel层的标准基生成了沿着Samuel起点的唯一的正则Rees代数，给出了代数和解析情形下强去奇异化的直接证明.
英文摘要：
Building upon ideas of Hironaka, Bierstone-Milman, Malgrange and others we generalize the inverse and implicit function theorem (in differential, analytic and algebraic setting) to sets of functions of larger multiplicities (or ideals). This allows one to describe singularities given by a finite set of generators or by ideals in a simpler form. In the special Cohen-Macaulay case we obtain a singular analog of the inverse function theorem. The singular implicit function theorem is closely related to a (proven here) extended version of the Weierstrass-Hironaka-Malgrange division and preparation theorems. The primary motivation for this paper comes from the desingularization problem. As an illustration of the techniques used, we give some applications of our theorems to desingularization extending some results on Hironaka normal flatness, the Samuel stratification and the Hilbert-Samuel function. The notion of the standard basis along Samuel stratum introduced in the paper (inspired by the Bierstone-Milman and Hironaka constructions) allows us to describe singularities along the Samuel stratum in a relatively simple way. It leads to a canonical reduction of the strong Hironaka desingularization with normally flat centers to a so called resolution of marked ideals. Moreover, in characteristic zero, the standard basis along Samuel stratum generates a unique canonical Rees algebra along Samuel startum giving a straightforward proof of the strong desingularization in algebraic and analytic cases.
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