Equisingularity Problems for Real Algebraic Singularities and Real Analytic Singularities

实代数奇点和实解析奇点的等奇性问题

基本信息

批准号：
18540084
负责人：
KOIKE Satoshi
金额：
$ 2.51万
依托单位：
Hyogo University of Teacher Education
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

项目摘要

A singular point is defined in Mathematics as a point at which the space is not smooth or a point at which the map is not regular. Equisingularity Problem is a problem to ask whether the singular points (resp. the families of singular points) of the spaces or maps are the same under some desirable equivalence relation (resp. triviality). It naturally becomes a problem there to introduce some equivalence relation, to ask if the equivalence relation is natural, to analyze the relation between the equivalence and the other equivalences, or to classify the singularities by the equivalence. In order to solve those problems, it is very important to establish the triviality theorem, to introduce some invariants, or to give characterizations for the equivalence. Concerning these things, I have got the following results as equisingularities of real algebraic singularities and real analytic singularities:(1) Let us consider the finiteness problem on some triviality for a family of zero-sets of Nash mappings defined over a not necessarily compact Nash manifold. The main results on this problem are:(i) Finiteness theorem holds on Blow-Nash triviality when the zero-sets have isolated singularities.(ii) Finiteness theorem holds on Blow-semialgebraic triviality in the non-isolated singularity case when the dimension of the zero-sets is 2 or 3.(iii) Finiteness theorem holds on the existence of Nash trivial simultaneous resolution without the assumptions on the isolated singularity or the dimension of the zero-sets.(2) I have got some necessary and sufficient conditions with Adam Parusinski for two variable real analytic functions to be blow-analytically equivalent. More precisely, two real analytic function germs of two variables are blow-analytically equivalent if and only if they have weakly isomorphic minimal resolutions, their real tree models are isomorphic, or they are cascade equivalent.

奇异点在数学中被定义为空间不平滑的点或地图不规则的点。等奇性问题是询问空间或映射的奇点（或奇点族）在某种理想的等价关系（或平凡性）下是否相同的问题。引入某种等价关系，询问等价关系是否自然，分析等价与其他等价之间的关系，或者通过等价对奇点进行分类，自然就成为一个问题。为了解决这些问题，建立平凡定理，引入一些不变量，或者给出等价的表征是非常重要的。关于这些事情，我得到了以下结果作为实代数奇点和实解析奇点的等奇异性：（1）让我们考虑在不一定紧凑的纳什上定义的零集纳什映射族的一些琐碎性的有限性问题歧管。该问题的主要结果是：(i) 当零集具有孤立奇点时，有限性定理在 Blow-Nash 平凡性上成立。(ii) 在非孤立奇点情况下，当维度为零集为 2 或 3。(iii) 有限性定理在纳什平凡联立解的存在性上成立，无需孤立奇点或假设(2) 我与 Adam Parusinski 一起得到了两个变量实解析函数吹解析等价的充分必要条件。更准确地说，两个变量的两个实解析函数萌芽是吹分析等价的，当且仅当它们具有弱同构的最小分辨率，它们的实树模型是同构的，或者它们是级联等价的。