Equisingularity Problems for Real Algebraic Singularities and Real Analytic Singularities

在数学中，奇点被定义为空间不光滑的点或映射不规则的点。等奇异性问题是一个关于奇异点是否存在的问题。空间或映射的奇点族在某些理想的等价关系下是相同的。琐事)。引入某个等价关系，问这个等价关系是否自然，分析这个等价和其他等价之间的关系，或者用等价来分类奇点，自然就成了一个问题。为了解决这些问题，建立琐屑定理，引入一些不变量，或者给出等价的刻画是非常重要的。关于这些问题，我得到了实代数奇点和实解析奇点的等奇异性：(1)我们考虑在不一定紧的纳什流形上定义的一组纳什映射的零集在一些琐碎上的有限性问题。该问题的主要结果是：(1)当零集具有孤立奇点时，Blow-Nash平凡性的有限性定理成立。（ii）当零集的维数为2或3时，在非孤立奇点情况下，有限定理成立blow -半代数平凡性。(3)有限定理证明了纳什平凡同时解析的存在性，而不需要对孤立奇点和零集维数的假设。(2)利用Adam Parusinski得到了两个变量实解析函数是吹解析等价的充分必要条件。更确切地说，两个变量的两个实解析函数胚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。

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.