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Research on invariants of real analytic singularities

实解析奇点不变量的研究

基本信息

批准号：
15540071
负责人：
KOIKE Satoshi
金额：
$ 2.3万
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：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

The aim of this research is to look for and introduce some invariants of real analytic singularities on topological equivalence, blow-analytic equivalence, Lipschitz equivalence, C^1 equivalence and so on, and to give classifications of real analytic singularities on those equivalences using the invariants and triviality theorems. Concerning these problems, I have obtained the following :1)In the joint work with Adam Parusinski, we have introduced motivic-type invariants as blow-analytic invariants for real analytic function germs, and have given blow-analytic classification of real analytic functions using our invariants and the Fukui invariants.2)I have worked with A.Parusinski also on a different type invariant from the above for real analytic functions of 2 variables. We have shown that the Newton boundary relative to an arc is a Lipschitz invariant and the Newton boundary with dots relative to an arc is a C^1 invariant in the 2 variables case.3)The Fukui invariant is well-known as … More a blow-analytic invariant for real analytic functioin germs. On the other hand, there is a problem whether the Fukui invariant is a topological invariant for complex holomorphic function germs. I have given the affirmative answer to it in the 2 variables case, and have constructed a negative example of 4 variables functions.4)For a semialgebraic mapping between semialgebraic sets, I and Mahiro Shiota have studied the set of points at which the fibre is not smooth. In particular, we have looked for an invariant for the triviality along the smooth part of a fibre. Then we have proved that a semialgebraic function is semialgebraically trivial along the smooth fibre, and constructed an example of a semialgebraic mapping which is not trivial along it.5)Related to the problem on Blow-Nash moduli for a family of algebraic singularities, I have proved with T.Fukui and K.Bekka that any polynomial mapping can be represented as the restriction of a desingularisation map of some algebraic variety to the intersection of the strict transform of the variety and the exceptional set at some point.6)I have introduced the notion of a sea-tangle neighborhood for a Lipschitz arc, and have shown that the degree and the width of the neighborhood are biLipschitz invariants and that the Briancon-Speder family and the Oka family are not biLpischitz trivial using the invariants. In addition, developing the idea, I have obtained a general result with Laurentiu Paunescu that the kissing dimension of subanalytic sets is a biLipschitz invariant. Less

本研究的目的是寻找并引入真实的解析奇点在拓扑等价、吹解析等价、Lipschitz等价、C^1等价等方面的一些不变量，并利用不变量和平凡性定理在这些等价上给出真实的解析奇点的分类。关于这些问题，我得到了以下结果：1）在与Adam Parusinski的合作工作中，我们引入了动机型不变量作为真实的解析函数芽的吹解析不变量，并使用我们的不变量和福井不变量给出了真实的解析函数的吹解析分类。2）我与A.Parusinski也研究了2个变量的真实的解析函数的不同类型的不变量。我们已经证明，在2个变量的情况下，相对于弧的牛顿边界是Lipschitz不变量，并且相对于弧的带有点的牛顿边界是C^1不变量。3）福井不变量众所周知为 ...更多信息真实的解析函数芽的吹解析不变量。另一方面，对于复全纯函数芽，福井不变量是否是拓扑不变量存在问题。我在二元情形下给出了肯定的回答，并构造了一个四维函数的反例。4）对于半代数集之间的半代数映射，我和Mahiro Shiota研究了纤维不光滑的点集。特别是，我们已经寻找了一个不变的平凡沿着光滑部分的纤维。然后证明了半代数函数沿着光滑纤维是半代数平凡的，并构造了一个沿光滑纤维不是平凡的半代数映射的例子。5）关于代数奇点族的Blow-Nash模问题，我与T.福井和K.贝卡一起证明了任何多项式映射都可以表示为某个代数簇的去奇异化映射对6）我已经引入了Lipschitz弧的sea-tangle邻域的概念，并且已经证明了邻域的度和宽度是biLipschitz不变量，并且Briancon-Speder家族和Oka家族不是biLpischitz平凡的。此外，发展的想法，我已经得到了一个一般结果与Laurentiu Paunescu接吻维数的次解析集是一个biLipschitz不变量。少