| Project/Area Number |
17540071
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Nagoya University |
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Principal Investigator |
SHIOTA Masahiro Nagoya University, Graduate School of Mathematics, professor, 大学院多元数理科学研究科, 教授 (00027385)
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| Co-Investigator(Kenkyū-buntansha) |
YASUMOTO Masahiro Nagoya University, Gratuate School of formation Science, professor, 大学院情報科学研究科, 教授 (10144114)
KOIKE Satoshi Hyogo University of Faculty of Teacher Education, School Education, professor, 学校教育学部, 教授 (60161832)
FUKUI Toshizumi Saitama University, Faculty of Science, professor, 理学部, 教授 (90218892)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2006: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2005: ¥2,000,000 (Direct Cost: ¥2,000,000)
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| Keywords | geometry / mathematical logic / algebra / 実代数幾何学 / モデル理論 / 半代数的集合 / 順序極小構造 / 定義可能集合 |
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| Research Abstract |
Functions with D-minimal structure and, especially, algebraic properties of analytic function ring were researched, and the following results were obtained, 1. By having invited Acquistapace and Broglia of Pisa University I investigated the family of global semianalytic sets. We tried to prove that the family is closed under the operations of taking finite union, finite intersection complement and cometed component. Then we solved the problem for dimension up to 4, and will publish in an article. 2. I visited University of Reines and worked jointly with caste on the problem of compactification of definable metric. In B-minimal structure there seems to be no differences between compactness and noncompactness. The problem concretizes this idea. We solved it and writed an article, which will be published. 3. J.Bolte of University of Paris 7 was invited, with who Lojasiewicz inequality was studied. The inequality was originally proved by Lojasiewicz in the local case. After then many mathematicians tried to generalized it. But any result was essentially local. We proved it globally and in any D-minimal structure. The work will appear in some journal. 4. Tu Le Loi of University of Dalat and I made joint researches on the ring of definable analytic functions. We concluded that there were no methods in research for general O-minimal structure, and adopted the hypothesis that the complexification of definable analytic function is definable. Then many important algebraic properties of the ring were shown, for example, noetherianness of the ring, Hilbert 17th problem and real Nallstellen Satz. We will write them in a paper or a book.
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