In this article, we functorially associate definable sets to $k$-analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of $X$ and show that they satisfy a bijective relation with the radial subsets of $X$. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of $k$-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser’s theorem on iso-definability of curves. However, our approach can also be applied to strictly $k$-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.

中文翻译：

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可定义的 BERKOVICH 曲线集

在本文中，我们在功能上将可定义集与$k$-解析曲线，以及它们之间解析态射的可定义映射，对于一大类$k$-解析曲线。给定一个$k$-解析曲线$X$，我们的关联允许我们有几个常见的 Berkovich 解析几何概念的可定义版本，例如从一个点发出的分支和在类型 2 的点处的剩余曲线。我们还描述了可定义对应物的可定义子集$X$并证明它们满足与径向子集的双射关系$X$. 作为一个应用程序，我们恢复（并略微扩展）Temkin 的结果，该结果涉及具有给定规定多重性的点集关于一个态射的径向性$k$-解析曲线。在分析代数曲线的情况下，我们的构造也可以看作是 Hrushovski 和 Loeser 关于曲线等可定义性的定理的显式版本。但是，我们的方法也可以严格地应用于$k$-仿射曲线和它们之间的任意态射，目前不在它们的设置范围内。