Research on real algebraic singularities

与复代数奇点的大量研究相比，真实的代数奇点的研究并不多。本课题的目的是研究后者。这里的真实的代数奇点是指Nash映射的零集的奇点。在本研究中，我得到了以下三个关于这些真实的代数奇点的结果：（I）我们说Nash集族允许blow-semi-algebraic tivialisation，如果存在Nash集族的同时分解和去奇异化Nash流形族的半代数平凡化，从而导致原Nash流形族的半代数平凡化。证明了一类二维Nash集的半平凡性的有限性定理。此外，还证明了一类三维真实的代数集的有限性定理。(II)令f：M → N是C^∞ Nash流形之间的一个C^∞ Nash映射，且dim M [大于或等于] 1，设K_κ表示f的纤维不是局部C^κ Nash流形的点集，其中κ= 1，2，.，∞.在与M.Shiota的合作工作中，我们证明了每一个K_k都是M的codim [大于或等于] 2的半代数集。此外，我们还证明了k_k是稳定的，即存在k ∈ N使得k_k = k_<k +1>=.= k_∞. (III)福井不变量是吹解析等价的不变量。一个Blow-Nash等价）的真实的解析函数芽（分别为. Nash function-germs）。在与S.Izumi和T. C. Kuo的合作工作中，我们给出了计算真实的和复杂情况下的福井不变量的公式，并描述了稳定性。此外，我们还阐明了福井不变量是二元复解析函数芽的拓扑不变量。

Comparing to an enormous number of researches on complex algebraic singularities, the number of researches on real algebraic singularities is not so large. The purpose of this project is to research the latter one. Here real algebraic singularities means singularities of the zero-sets of Nash mappings. In this research, I have got the following three results concerning these real algebraic singularities :(I) We say that the family of Nash sets admits a blow-semialgebraic tivialisation, if there are a simultaneous resolution of the family of Nash sets and a semialgebraic trivialisation of the family of the desingularised Nash manifolds which induces a semialgebraic one of the original family. I proved a finiteness theorem on semilagebraic triviality for a family of 2-dimensional Nash sets. In addition, I proved a finiteness theorem on it for a family of 3-dimensional real algebraic sets.(II) Let f : M → N be a C^∞ Nash mapping between C^∞ Nash manifolds with dim M 【greater than or equal】 1, and let Σ_κ denote the set of points at which the fiber of f is not a locally C^κ Nash manifold for κ=1,2, …, ∞. In the joint work with M.Shiota, we proved that each Σ_κ is a semialgebraic set of M of codim 【greater than or equal】 2. In addition, we showed that Σ_κ is stabilised, namely, there is κ∈ N such that Σ_κ=Σ_<κ+1>=…=Σ_∞.(III) The Fukui invariant ie well-known as an invariant for a blow-analytic equivalence (resp. a blow-Nash equivalence) of real analytic function-germs (resp. Nash function-germs). In the joint work with S.Izumi and T.C.Kuo, we gave a formula to compute the Fukui invariant in the real and complex cases, and we characterised the stability. In addition, we clarified that the Fukui invariant is a topological invariant for 2-variables complex analytic function-germs.