论文标题
在尺寸，标准零件地图和$ p $的封闭字段上
On Dimensions, Standard Part Maps, and $p$-Adically Closed Fields
论文作者
论文摘要
本文的目的是研究$ p $ -Adic数字$ {{\ Mathbb Q} _p} $及其基本扩展$ K $的尺寸和标准零件图。我们表明，对于任何$ K $ -DEFINABLE，$ x \ subseteq k^m $，$ \ text {dim} _k（x）\ geq \ geq \ text {dim} _ {{\ mathbb q} __p} _令$ v \ subseteq k $为$ k $ $ k $ of $ {{\ mathbb q} _p} $和$ \ text {\ st}的凸壳：v \ rightarrow {{\ mathbb q} _p} _p} _P} _是标准零件映射。我们表明，对于任何$ k $ -ddable的函数$ f：k^m \ rightarrow k $，都有可定义的子集$ d \ subseteq {{\ mathbb q} _p} _p}^m $，以至于$ {{\ mathbb q} _pp} _ _} _ _} _ m \ backslash d $ in Interion unterion $ in $ in $ in $ v $和$ \ text {st}（f（\ text {st}^{ - 1}（x）））$是常数，或$ f（\ text {st}^{ - 1} { - 1}（x）（x））\ cap v = \ emberySet $。我们还证明，$ \ text {dim} _k（x）\ geq \ text {dim} _ {{\ mathbb q} _p} _p}（\ text {st}（x \ cap v^m）$对于每个定义的$ x \ subseteq k^m $。
The aim of this paper is to study the dimensions and standard part maps between the field of $p$-adic numbers ${{\mathbb Q}_p}$ and its elementary extension $K$ in the language of rings $L_r$. We show that for any $K$-definable set $X\subseteq K^m$, $\text{dim}_K(X)\geq \text{dim}_{{\mathbb Q}_p}(X\cap {{\mathbb Q}_p}^m)$. Let $V\subseteq K$ be convex hull of $K$ over ${{\mathbb Q}_p}$, and $\text{\st}: V\rightarrow {{\mathbb Q}_p}$ be the standard part map. We show that for any $K$-definable function $f:K^m\rightarrow K$, there is definable subset $D\subseteq{{\mathbb Q}_p}^m$ such that ${{\mathbb Q}_p}^m\backslash D$ has no interior, and for all $x\in D$, either $f(x)\in V$ and $\text{st}(f(\text{st}^{-1}(x)))$ is constant, or $f(\text{st}^{-1}(x))\cap V=\emptyset$. We also prove that $\text{dim}_K(X)\geq \text{dim}_{{\mathbb Q}_p}(\text{st}(X\cap V^m))$ for every definable $X\subseteq K^m$.
