关于p-进闭域的非标准模型上的维数和标准映射

On Dimensions,Standard Part Maps,and p-Adically Closed Fields

我们研究了p-进数域Qp的初等扩张K上的可定义函数及其相关的维数理论。证明了Km的可定义子集X在标准模型Qp中的维数dimQp(X∩Qpm)不超过X在K中的维数。我们还证明了:对K中的可定义函数f:Km→K,存在一个可定义集合D■Qpm以及Qp上的可定义函数g:D→Qp∪{∞}使得Qpm \D无内点,且对每个x∈D有g(x)=st(f(st-1(x)))。进一步,我们证明了对Km的可定义子集X,有dimQp(st(X))也不超过X在K中的维数。

The aim of this paper is to study the dimensions and standard part maps between the field of p-adic numbers Qp and its elementary extension K in the language of rings Lr. We show that for any K-definable set X ■ Km, dim K(X) ≥ dimQp(X ∩ Qpm). Let V ■ K be convex hull of K over Qp, and st : V → Qp be the standard part map. We show that for any K-definable function f : Km→ K, there is definable subset D ■ Qpm such that Qpm \D has no interior, and for all x ∈ D, either f(x) ∈ V and st(f(st-1(x))) is constant, or f(st-1(x)) ∩ V = ■. We also prove that dimK(X) ≥ dimQp(st(X ∩ Vm)) for every definable X ■ Km.