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Attach a (Laurent) monomial weight $x_1^{i_1} \cdots x_n^{i_n}$ to each point $(i_1, \dots, i_n)$ of $\mathbb{Z}^{n}$ and call it $\mathbb{Z}^{n}[x_1, x_{1}^{-1}, \dots, x_n, x_n^{-1}]$. Does this formal space have a name (other than the title) or any special properties?

I understand that Newton polygons are studied using this space. How else is it useful?

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It is the group algebra of the group $\mathbb Z^n$, so it shows up in lots of places. One place where you see it a lot is in toric geometry.

What exactly do you want to know?

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    I'm just looking for a general reference where I can read about these spaces and how they are used. I'm working with some generating functions which seem to count objects in this space.2011-03-10