arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2002.10117math.LO
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关于尺寸、标准零件图和$p$ --基本闭域

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

Ningyuan Yao

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摘要：

本文的目的是研究环语言$L_r$中$p$ -进数域${{\mathbb Q}_p}$与其初等扩张$K$之间的维数和标准部分映射.我们证明了对于任何$K$可定义集$X\subseteq K^m$，$\text{dim}_K(X)\geq \text{dim}_{{\mathbb Q}_p}(X\cap {{\mathbb Q}_p}^m)$ .设$V\subseteq K$为$K$在${{\mathbb Q}_p}$上的凸包，$\text{\st}: V\rightarrow {{\mathbb Q}_p}$为标准零件图.我们表明，对于任何$K$可定义的功能$f:K^m\rightarrow K$，有可定义的子集$D\subseteq{{\mathbb Q}_p}^m$，${{\mathbb Q}_p}^m\backslash D$没有内部，并为所有$x\in D$，无论是$f(x)\in V$和$\text{st}(f(\text{st}^{-1}(x)))$是常数，或$f(\text{st}^{-1}(x))\cap V=\emptyset$。我们还证明了对于每个可定义的$X\subseteq K^m$，$\text{dim}_K(X)\geq \text{dim}_{{\mathbb Q}_p}(\text{st}(X\cap V^m))$。

英文摘要：

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 $X\subseteq K^m$.

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