Some local definability theory for holomorphic functions

Some local definability theory for holomorphic functions
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全纯函数的一些局部可定义理论

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发表时间：
2008
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通讯作者：
A. Wilkie
A. Wilkie
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作者：
A. Wilkie

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设Mathcal{F}是全纯函数的集合，Mathbb{R}(PR(Mathcal{F})) 表示结构mathbb{R}_{an}到有序域运算的约化 连同一组适当的限制(见下文) 数学{F}中所有函数的实部和虚部。我们问这个问题： 哪些全纯函数是局部可定义的(即有实数 和可局部定义的虚部)在结构mathbb{R}(PR(mathcal{F}))中？它 很容易看到，所有这类函数的集合都关闭在 组合、偏微分、隐式可定义性(通过 一因变量隐函数定理)和Schwarz 倒影。我们推测这用尽了所有的可能性，而我们 在通用点附近也证明了这一点。更准确地说， 我们证明了这四种运算决定了自然的准几何 与Mathbb{R}(PR(数学{F}))-可定义的全纯函数相关联。
Let mathcal{F} be a collection of holomorphic functions and let mathbb{R}(PR(mathcal{F})) denote the reduct of the structure mathbb{R}_{an} to the ordered field operations together with the set of proper restrictions (see below) of the real and imaginary parts of all functions in mathcal{F}. We ask the question: Which holomorphic functions are locally definable (ie have their real and imaginary parts locally definable) in the structure mathbb{R}(PR(mathcal{F}))? It is easy to see that the collection of all such functions is closed under composition, partial differentiation, implicit definability (via the Implicit Function Theorem in one dependent variable) and Schwarz Reflection. We conjecture that this exhausts the possibilities and we prove as much in the neighbourhood of generic points. More precisely, we show that these four operations determine the natural pregeometry associated with mathbb{R}(PR(mathcal{F}))-definable, holomorphic functions.