Equisingularity Problems for Real Algebraic Singularities and Real Analytic Singularities

奇点在数学中被定义为空间不光滑的点或映射不规则的点。均衡性问题是问奇异点(对应于。在某种理想的等价关系下，空间或映射的奇点族是相同的。琐碎)。引入某种等价关系，问这种等价关系是否是自然的，分析等价与其他等价之间的关系，或者根据等价对奇点进行分类，自然成为一个问题。为了解决这些问题，建立平凡定理、引入一些不变量或给出等价的刻画是非常重要的。关于这些问题，我得到了如下结果，作为实代数奇点和实解析奇点的等价性：(1)让我们考虑定义在不一定紧的Nash流形上的零集族Nash映射族的平凡有限问题。关于这个问题的主要结果是：(I)当零点集具有孤立奇点时，有限定理成立.(Ii)当零点集的维度为2或3时，有限定理在非孤立奇点情况下成立.(Iii)有限定理成立于Nash平凡平凡同时归结的存在性，不需要假设零点的孤立奇点或零点集的维度.(2)我与Adam Parusinski得到了两个变量实解析函数是爆破解析等价的一些充要条件.更确切地说，两个二元实解析函数芽吹解析等价的充要条件是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.