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Research of real algebraic geometry by model theory

模型论研究实代数几何

基本信息

批准号：
17540071
负责人：
SHIOTA Masahiro
金额：
$ 2.11万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

项目摘要

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.

研究了具有D-极小结构的函数，特别是解析函数环的代数性质，得到了如下结果：1。通过邀请Acquistapace和Broglia的比萨大学我调查了家庭的全球半解析集。我们试图证明这个族在取有限并、有限交补和余分支的运算下是封闭的。然后，我们解决了维度高达4的问题，并将发表在一篇文章中。2.我访问了大学的雷恩和共同工作与种姓的问题紧的可定义度量。在B-极小结构中，紧性和非紧性似乎没有区别。这个问题使这个想法具体化。我们解决了这个问题，并写了一篇文章，将发表。3.邀请了巴黎第七大学的J.Bolte，与他一起研究了Lojasiewicz不等式。这个不等式最初是由Lojasiewicz在局部情况下证明的。此后，许多数学家试图将其推广，但任何结果本质上都是局部的。我们证明了它在全球范围内，并在任何D-最小结构。这项工作将出现在某个期刊上。4.大叻大学的Tu Le Loi和我共同研究了可定义解析函数的环。我们认为目前还没有研究一般O-极小结构的方法，并采用了可定义解析函数的复化是可定义的假设。然后给出了环的许多重要的代数性质，如Noether性，Hilbert第17问题和真实的Nallstelen Satz等.我们会把它们写在纸上或书上。