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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 $k[T^{d}]$?

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    $\begingroup$ Yes. This is Borel - Linear algebraic groups, Proposition 8.4. $\endgroup$
    – LSpice
    Commented Apr 22, 2023 at 1:25
  • $\begingroup$ this proposition says that a diagonalizable group is isomorphic to a subgroup of diagonal matrices. I already know it is a torus, how it is relevant? $\endgroup$
    – prochet
    Commented Apr 22, 2023 at 6:34
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    $\begingroup$ Re, it also says that a subgroup of diagonal matrices, like yours, is diagonalisable. The definition of diagonalisability (see the beginning of Section 8.2) is that the character lattice spans the ring of regular functions. $\endgroup$
    – LSpice
    Commented Apr 22, 2023 at 6:40

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Yes, more precisely, $k[T^d]$ is the group algebra on $\mathbb{Z}^d$. See Milne, Algebraic Groups, 12.3.

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