Research on real algebraic singularities

与复代数奇点的大量研究相比，实代数奇点的研究并不是很多。本课题的目的是对后者进行研究。这里，实代数奇点指的是纳什映射的零集的奇点。在这项研究中，我得到了关于这些实代数奇点的三个结果：(I)如果同时存在Nash集族的分解和非奇异Nash流形族的半代数平凡化，则我们说Nash集族允许Blow-半代数平凡化，从而诱导出原始族的半代数流形。证明了关于二维Nash集族的半代数平凡性的一个有限定理。(2)设f：M→N是C^∞纳什流形之间的C^∞纳什映射，其中DIMM[大于等于]1，Σ_κ表示f的纤维不是局部C^κ纳什流形且κ=1，2，…的点集、∞.在与M.Shiota的合作中，我们证明了每个Σ_κ都是码[大于或等于]2的M的半代数集。此外，我们还证明了Σ_κ是稳定的，即存在κ∈N使得Σ_κ=Σ_&lt；κ+1&gt；=…=Σ_∞。(Iii)Fukui不变量，即众所周知的吹扫解析等价的不变量。实解析函数芽的一个Blow-Nash等价)。Nash函数-细菌)。在与S.Izumi和T.C.Kuo的合作中，我们给出了实数和复数情形下Fukui不变量的计算公式，并刻画了稳定性。此外，我们还阐明了Fukui不变量是二元复解析函数芽的一个拓扑不变量。

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.