Research on real algebraic singularities

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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.