Let $E\subseteq {\mathbb{Q}}_p^n$ and T be a set‐valued map from E to ${\mathbb{Q}}_p^m$. We prove that if T is p‐adic semi‐algebraic, lower semi‐continuous and $T(x)$ is closed for every $x\in E$, then T has a p‐adic semi‐algebraic continuous selection. In addition, we include three applications of this result. The first one is related to Fefferman's and Kollár's question on existence of p‐adic semi‐algebraic continuous solution of linear equations with polynomial coefficients. The second one is about the existence of p‐adic semi‐algebraic continuous extensions of continuous functions. The other application is on the characterization of right invertible p‐adic semi‐algebraic continuous functions under the composition.

中文翻译：

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关于p-adic半代数连续选择

让 $Ë\subseteq {\mathbb{问}}_p^{ñ}$和T是从E到${\mathbb{问}}_p^{米}$。我们证明如果T是p -adic半代数，则下半连续和$Ť（X）$ 每个人都关闭 $X\in Ë$，则T具有p -adic半代数连续选择。此外，我们包括此结果的三个应用程序。第一个与Fefferman和Kollár有关具有多项式系数的线性方程组的p -adic半代数连续解的存在性问题有关。第二个是关于连续函数的p -adic半代数连续扩展的存在。另一个应用是在合成下表征右可逆p -adic半代数连续函数。