Let $T={\mathbb{G}}_{m}^{r}$ be the split multiplicative torus of rank $r$ with coordinates $z_{1},\dots,z_{r}$. Then a point $x$ of ${T^{\rm an}}$ could be visualized by the coordinates $z_{1}(x),\dots,z_{r}(x)\in{\mathscr{H}}(x)$ and the multiplicative seminorm corresponding to $x$ is given by $|f(x)|=|f(z_{1}(x),\dots,z_{r}(x))|$ for every Laurent polynomial $f$ on $T$. Conversely, every field extension $L/K$ with an absolute value extending the given absolute value on $K$ and every $(\beta_{1},\dots,\beta_{r})\in L^{r}$ give rise to a point $x\in{T^{\rm an}}$ by $|f(x)|:=|f(\beta_{1},\dots,\beta_{r})|$. Note that $L$ and $(\beta_{1},\dots,\beta_{r})$ are not uniquely determined by $x$.

In particular, we get an inclusion of $T(K)$ into ${T^{\rm an}}$. For every $x\in T(K)$, we have $d(x)=0$. However, there can be also other points with $d(x)=0$. If $T={\mathbb{G}}_{m}^{1}$, then precisely the points of type 1 (i.e. the $K$-rational points) and the points of type 4 satisfy $d(x)=0$ (see [Be90], 1.4.4).

Returning to the case $T={\mathbb{G}}_{m}^{r}$, there are some distinguished points of ${T^{\rm an}}$ which behave completely different than $K$-rational points. For positive real numbers $s_{1},\dots,s_{r}$, we define the associated weighted Gauss norm on $K[T]$ by

for every Laurent polynomial $f=\sum_{{\mathbf{m}}\in{\mathbb{Z}}^{r}}\alpha_{\mathbf{m}}{\mathbf{z}}^{\mathbf{m}}\in K[T]=K[z_{1}^{\pm 1},\dots,z_{r}^{\pm 1}]$. It follows from the Gauss Lemma that the weighted Gauss norm is a multiplicative seminorm giving rise to a point $\eta_{\mathbf{s}}\in{T^{\rm an}}$. The set $S({T^{\rm an}}):=\{\eta_{\mathbf{s}}\mid s_{1}>0,\dots,s_{r}>0\}$ is called the skeleton of ${T^{\rm an}}$. Every point $\eta_{\mathbf{s}}\in S({T^{\rm an}})$ satisfies $d(\eta_{\mathbf{s}})=r$ (see [Du12], (0.12) and (0.13)).