Let $T$ be an algebraic torus over an algebraically closed field $k$. Let $d\in\mathbb{N}^{*}$. For every $d$-tuple of integers $\underline{n}=(n_1,\dots, n_d)$ and a function $f\in k[T]$, we can consider the functions $f_{\underline{n}}$ on $T^{d}$ given by:

$(t_1,\dots, t_d)\mapsto f(t_1^{n_{1}}\dots t_{d}^{n_{d}})$.

Do these functions generate $k[T^{d}]$?

Let $T$ be an algebraic torus over an algebraically closed field $k$. Let $d\in\mathbb{N}^{*}$. For every $d$-tuple of integers $\underline{n}=(n_1,\dotsc, n_d)$ and a function $f\in k[T]$, we can consider the functions $f_{\underline{n}}$ on $T^{d}$ given by:

$$(t_1,\dotsc, t_d)\mapsto f(t_1^{n_{1}}\dotsm t_{d}^{n_{d}}).$$

Do these functions generate $k[T^{d}]$?