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The n-dimensional algebraic torus is defined as $ \mathbb T^n ={(a_1,a_2,...,a_n): a_i \in K^*} $.

What is the motivation behind this definition?

And why affine varieties in torus is defined using Laurent polynomials rather than ordinary polynomials?

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    Yes. I me$a$n motiv$a$tion2011-12-28

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One reason that such varieties are called "tori" is that they play the same role in the theory of algebraic groups over $K$ that compact tori (products of circles, $(S^1)^n$) play in the theory of compact Lie groups.

Indeed, the complex torus $(\mathbb{C}^\times)^n$ is the complexification of the compact torus $(S^1)^n$, as follows. The compact torus can be written as the real variety $T=\{(x_1,y_1,\ldots,x_n,y_n)\in\mathbb{R}^{2n}: x_k^2+y_k^2=1, 1\leq k\leq n\}.$

The complexification of this is

$T_\mathbb{C}\{(x_1,y_1,\ldots,x_n,y_n)\in\mathbb{C}^{2n}: x_k^2+y_k^2=1, 1\leq k\leq n\}.$

But now we can make the change of coordinates $u_k = x_k+iy_k$, $v_k = x_k-iy_k$, and obtain $T_\mathbb{C}\cong\{(u_1,v_1,\ldots,u_n,v_n)\in\mathbb{C}^{2n}: u_kv_k=1\}.$

This is identified with $(\mathbb{C}^\times)^n$ by taking the coordinates $u_1,\ldots,u_n$.

As to why affine subvarieties of the torus are defined using Laurent polynomials, this is because the Laurent polynomials are the regular functions on the torus. As above, we can realize $\mathbb{T}^n$ as the affine variety $\mathbb{T}^n = \{(a_1,b_1,\ldots,a_n,b_n)\in K^{2n}: a_kb_k=1\},$ and the polynomial functions on $K^n$ restrict to Laurent polynomials on $\mathbb{T}^n$ since $b_k = a_k^{-1}$.