Expansions of Algebraically closed Fields II: Functions of several Variables
Expansions of Algebraically closed Fields II: Functions of several Variables
复制标题
代数闭域的展开 II：多变量函数
DOI：
10.1142/s0219061303000224
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
S. Starchenko
中科院分区：
文献类型：
--
作者：
Y. Peterzil;S. Starchenko
令 ℛ 为实闭域 R 的 o 最小展开。我们在这里继续我们在[11]中开始的关于代数闭域可微性的研究。我们发展了多变量可定义函数的 K-可微性的基本理论，证明了可去除奇点的定理以及可定义函数的 Weierstrass 准备和除法定理的类似物。我们还考虑可定义的亚纯函数，并证明每个在 Kn 上亚纯的可定义函数必然是有理函数。最后，我们讨论了复解析流形的可定义类似物，以及与紧复流形上的模型理论工作的可能联系，并在我们的环境中提出了两个“非标准流形”的例子。
Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field . We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is meromorphic on Kn is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting.