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。等奇异性问题是一个问题，问是否奇异点（分别。奇异点族）在某种理想的等价关系下（分别为琐碎）。在那里，引入某种等价关系，询问这种等价关系是否自然，分析这种等价关系与其他等价关系之间的关系，或者根据等价关系对奇点进行分类，自然就成了一个问题。为了解决这些问题，建立平凡性定理，引入一些不变量，或给出等价的特征是非常重要的。关于这些问题，我得到了下列结果作为真实的代数奇点和真实的解析奇点的等奇异性：（1）考虑定义在不一定紧的Nash流形上的Nash映射的零集族在平凡性上的有限性问题。主要结果是：（1）当零集有孤立奇点时，Blow-Nash平凡性的显著性定理成立。(ii)当零集的维数为2或3时，在非孤立奇点情形下，Blow-半代数平凡性的单调性定理成立。(iii)在没有孤立奇点和零集维数的条件下，证明了Nash平凡同时分解的存在性。(2)我和Adam Parusinski一起得到了二元真实的解析函数blow-analytical等价的一些充分必要条件。更精确地说，两个二元的真实的解析函数芽是吹解析等价的当且仅当它们具有弱同构的极小分解，它们的真实的树模型是同构的，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.