奇点局部奇点的分解的构造性初等方法
A constructive elementary method for local resolution of singularities
摘要：
本文对文献[G1]-[G3]中的局部奇点的分解算法进行了简化和改进，给出了一种局部奇点的分解方法，它适用于特征为零的任意局部域上具有收敛幂级数的函数.本文的算法完全是一个初等的经典分析论证，只利用了隐函数定理和关于幂级数及牛顿多面体的基本事实。给出了几个例子。这些方法与传统的奇点的分解方法有很大的不同。在另一篇论文arxiv：1104.4684中，本文的方法（但大部分不是奇点的奇点的分解定理本身）被用来证明关于振荡积分、指数和及相关问题的结果。注意事项：这篇论文应该被看作是旧版本arxiv：1104.4684中奇点的奇点的分解部分的替代，因为这篇论文与那个算法有很大的不同，应该更容易阅读。首先，这里使用的是标准的blowup，而不是arxiv：1104.4684中的单项映射，后者迫使人们处理与这种单项映射不一定是一对一相关的技术问题。其次，arxiv：1104.4684中使用牛顿多面体进行的复杂迭代域细分已被简化为第2节中给出的简短一步论证。第三，在新算法的上下文中，形成单位的划分可以更自然地完成（见引理3.5）。
英文摘要：
In this paper we simplify and otherwise improve the local resolution of singularities algorithm of [G1]-[G3], providing a local resolution of singularities method that works for functions with convergent power series over an arbitrary local field of characteristic zero. The algorithm of this paper is an entirely elementary classical analysis argument, using only the implicit function theorem and elementary facts about power series and Newton polyhedra. Several examples are given. The methods are quite different from traditional resolution of singularities methods. In a separate paper arxiv:1104.4684, the methods of this paper (but for the most part not the resolution of singularities theorems themselves) are used to prove results concerning oscillatory integrals, exponential sums, and related matters. Note: This paper should be viewed as a replacement for the resolution of singularities portion of older versions of arxiv:1104.4684, as this paper is quite a bit different from that algorithm, and should be a lot easier to read. For one, standard blowups are used here, rather than the monomial maps of arxiv:1104.4684 which forced one to deal with technical issues related to such monomial maps not necessarily being one to one. Secondly, the complicated iterated domain subdivisions using Newton polyhedra that were done in arxiv:1104.4684 have been reduced to a short one-stage argument, given in section 2. Thirdly, forming partitions of unity can be done more naturally in the context of the newer algorithm (see Lemma 3.5).
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