Project/Area Number  10640075 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Geometry

Research Institution  Hyogo University of Teacher Education 
Principal Investigator 
KOIKE Satoshi Hyogo University of Teacher Education, Faculty of School Education, Associate Professor, 学校教育学部, 助教授 (60161832)

CoInvestigator(Kenkyūbuntansha) 
FUKUI Toshizumi Saitama University, Faculty of Science, Associate Professor, 理学部, 助教授 (90218892)
SHIOTA Masahiro Nagoya University, Garduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (00027385)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 2000 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1999 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1998 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  modified Nash triviality / blowanalyticity / Nash approximation theorem / isotopy lemma / toric desingularization / Fukui invariant / flatness / complexity / モディファイド・ナッシュ自明性 / ブロー解析性 / ナッシュ近似定理 / イソトピー補題 / トーリック特異点解消 / 福井不変量 / 平坦性 / 複雑性 / トーソック特異点解消 
Research Abstract 
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 zerosets 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 blowsemialgebraic 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 2dimensional Nash sets. In addition, I proved a finiteness theorem on it for a family of 3dimensional 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 wellknown as an invariant for a blowanalytic equivalence (resp. a blowNash equivalence) of real analytic functiongerms (resp. Nash functiongerms). 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 2variables complex analytic functiongerms.
