论文标题
$ P $ - 最小字段和应用程序的可确定性
Definable completeness of $P$-minimal fields and applications
论文作者
论文摘要
我们表明，$ p $ - 最小的field $ k $的每个可确定的封闭和有限子集的嵌套家族都有非空交叉路口。作为一个应用程序，我们回答了一个darnière和halupczok的问题，表明$ p $ - 最小字段满足“极值属性”：对于每个封闭且有限的子集$ \ u \ subseteq k $以及每个可解释的连续函数$ f \ colon u \ colon u \toγ_k$（其中$γ_k$ debal the $γ_k$ debal the $γ_k$ exites $ f（ud）由于他们的工作，获得了另外两个推论。第一个表明，$ k \timesγ_k^n $的每个可解释子集已经在戒指的语言中解释，回答了一个cluck和halupczok的问题。这特别意味着每个$ p $ - 最小的字段都具有多元界限。第二个以满足经典细胞制备定理的$ p $少量字段为特征的，将其定为具有确定的Skolem功能的定理，从而推广了哀悼的结果。
We show that every definable nested family of closed and bounded subsets of a $P$-minimal field $K$ has non-empty intersection. As an application we answer a question of Darnière and Halupczok showing that $P$-minimal fields satisfy the "extreme value property": for every closed and bounded subset $U\subseteq K$ and every interpretable continuous function $f\colon U \to Γ_K$ (where $Γ_K$ denotes the value group), $f(U)$ admits a maximal value. Two further corollaries are obtained as a consequence of their work. The first one shows that every interpretable subset of $K\timesΓ_K^n$ is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every $P$-minimal field is polynomially bounded. The second one characterizes those $P$-minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues.
