We provide a direct proof of a Bogomolov-type statement for affine varieties V defined over function fields K of finite transcendence degree over an arbitrary field k, generalising a previous result (obtained through a different approach) of the first author in the special case when K is a function field of transcendence degree $1$ . Furthermore, we obtain sharp lower bounds for the Weil height of the points in $V(\overline {K})$ , which are not contained in the largest subvariety $W\subseteq V$ defined over the constant field $\overline {k}$ .

中文翻译：

---

在有限超越度的函数域上定义的仿射变体上的小高度点

我们为仿射品种提供 Bogomolov 型陈述的直接证明五在函数字段上定义ķ任意域上的有限超越度ķ，在特殊情况下概括第一作者的先前结果（通过不同的方法获得）ķ是超越度的函数场$1$. 此外，我们获得了点的 Weil 高度的急剧下界$V(\overline {K})$, 不包含在最大的子品种中$W\子集 V$在常量字段上定义$\overline {k}$.