Let $0<m_{1}<\dots<m_{r}$ be integer numbers with $\gcd(m_{1},\dots,m_{r})=1$, and $p_{1},\dots,p_{r}\in K^{\times}$. Let $\varphi\colon\mathbb{T}\to\mathbb{P}^{r}$ be the map given by $\varphi(t)=(1:p_{1}t^{m_{1}}:\dots:p_{r}t^{m_{r}})$ and let $Y$ be the closure of the image of $\varphi$. Consider the polynomial $q\in K[z]$ defined as

$\displaystyle q=\begin{cases}1+\sum_{j=1}^{r}|p_{j}|^{2}z^{m_{j}},&\text{ in the Archimedean case},\\ 1+\sum_{j=1}^{r}p_{j}z^{m_{j}},&\text{ in the non-Archimedean case}.\end{cases}$

Let $\{\xi_{i}\}_{i}\subset{\overline{K}}^{\times}$ be the set of roots of $q$ and, for each $i$, let $\ell_{i}\in\mathbb{N}$ be the multiplicity of $\xi_{i}$. Let ${\overline{L}}$ and $s$ be as in Proposition 8.1. Then, in the Archimedean case,

1. (1)

   $\displaystyle\psi_{{\overline{L}},s}(u)=-\log|p_{r}|-\frac{1}{2}\sum_{i}\ell_{i}\log|\operatorname{e}^{-2u}-\xi_{i}|$ for $u\in\mathbb{R}$,

2. (2)

   $\displaystyle{\mathcal{M}}_{\mathbb{Z}}(\psi_{{\overline{L}},s})=-2\sum_{i}\ell_{i}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})^{2}}\,\,\text{\rm d}u$,

3. (3)

   $\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\frac{1}{2}\sum_{i}\ell_{i}^{2}+\frac{1}{2}\sum_{i<j}\ell_{i}\ell_{j}\frac{\xi_{i}+\xi_{j}}{\xi_{i}-\xi_{j}}(\log(-\xi_{i})-\log(-\xi_{j}))$, where $\log$ is the principal determination of the logarithm.

While in the non-Archimedean case,

1. (4)

   $\displaystyle\psi_{{\overline{L}},s}(u)={\operatorname{val}}_{K}(p_{r})+\sum_{i}\ell_{i}\min\{u,{\operatorname{val}}_{{\overline{K}}}(\xi_{i})\}$ for $u\in\mathbb{R}$,

2. (5)

   $\displaystyle{\mathcal{M}}_{\mathbb{Z}}(\psi_{{\overline{L}},s})=\sum_{i}\ell_{i}\delta_{{\operatorname{val}}_{{\overline{K}}}(\xi_{i})}$,

3. (6)

   $\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\sum_{i<j}\ell_{i}\ell_{j}\log(\max\{1,|\xi_{i}|/|\xi_{j}|\})$.