论文标题
Delon和Yao的P-ADIC结果的nippy证明
Nippy proofs of p-adic results of Delon and Yao
论文作者
论文摘要
让$ k $是$ \ mathbb {q} _p $，$ v $的基本扩展，是k $中的有限$ a \，$ \ mathrm {st} $是标准零件映射$ k^m \ to \ mathbb {q} Delon表明$ \ Mathbb {q}^m_p \ cap x $ is $ \ mathbb {q} _p $ -definable。 Yao表明，$ \ dim \ mathbb {q}^m_p \ cap x \ leq x \ leq x $ and $ \ dim \ mathrm {st}（v^n \ cap x）\ leq \ leq \ leq \ dim x $。我们给出了新的$ \ mathrm {nip} $ - 这些结果的理论证明，并表明两者在更一般的环境中都存在。我们还证明了扩展的类似结果，$ \ mathbb {q}^{\ mathrm {an}} _ p $ of $ \ mathbb {q} _p $ y Mathbb {q} _p $ y y Mathbb {z}作为一个应用程序，我们表明，如果$（x_k）_ {k \ in \ mathbb {n}} $是$ \ mathbb {q}^{Q}^{\ mathrm {an}} _ p $ - 可定义的子集的$ \ mathbb $ \ mathbb intogge utgenge in poptoly的$ \ mathbb {q}^{ \ subseteq \ mathbb {q}^m_p $，然后$ x $是$ \ mathbb {q}^{\ mathrm {an}} _ p $ -definable和$ \ dim x \ dim x \ leq \ leq \ limsup_ {
Let $K$ be an elementary extension of $\mathbb{Q}_p$, $V$ be the set of finite $a \in K$, $\mathrm{st}$ be the standard part map $K^m \to \mathbb{Q}^m_p$, and $X \subseteq K^m$ be $K$-definable. Delon has shown that $\mathbb{Q}^m_p \cap X$ is $\mathbb{Q}_p$-definable. Yao has shown that $\dim \mathbb{Q}^m_p \cap X \leq \dim X$ and $\dim \mathrm{st}(V^n \cap X) \leq \dim X$. We give new $\mathrm{NIP}$-theoretic proofs of these results and show that both inequalities hold in much more general settings. We also prove the analogous results for the expansion $\mathbb{Q}^{\mathrm{an}}_p$ of $\mathbb{Q}_p$ by all analytic functions $\mathbb{Z}^n_p \to \mathbb{Q}_p$. As an application we show that if $(X_k)_{k \in \mathbb{N}}$ is a sequence of elements of an $\mathbb{Q}^{\mathrm{an}}_p$-definable family of subsets of $\mathbb{Q}^m_p$ which converges in the Hausdroff topology to $X \subseteq \mathbb{Q}^m_p$ then $X$ is $\mathbb{Q}^{\mathrm{an}}_p$-definable and $\dim X \leq \limsup_{k \to \infty} \dim X_k$.
