Research of real algebraic geometry by model theory

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.通过邀请比萨大学的Acquistapace和Broglia，我研究了整体半解析集族。证明了在取有限并、有限交补和合成分支的运算下，该族是封闭的。然后我们解决了4维以下的问题，并将发表在一篇文章中。2.我访问了莱因斯大学，并与卡斯特共同研究了可定义度量的紧凑化问题。在B-极小结构中，紧性和非紧性似乎没有区别。这个问题将这一想法具体化。我们解决了这个问题，写了一篇文章，并将发表。3.邀请了巴黎7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在当地的案例中证明的。此后，许多数学家试图对其进行推广。但任何结果基本上都是地方性的。我们在任何D-极小结构中全局地证明了它。这项研究将发表在一些期刊上。4.我和达拉特大学的屠乐来共同研究了可定义解析函数环。我们得出结论，一般的O-极小结构的研究没有方法，并且采用了可定义解析函数的复化是可定义的假设。然后证明了环的许多重要的代数性质，如环的无以太性、Hilbert第17问题和实Nallstellen Satz。我们会把它们写在一张纸或一本书上。

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.