Let $T=(\mathbb{C}^{\times})^n$ be the $n$-dimensional torus acting on the polynomial algebra $\mathbb{C}[x_1,x_2, \ldots,x_n]$ diagonally, i.e. $$ diag(t^{a_1},t^{a_2},\ldots,t^{a_n})x_i=t^{a_i}x_i, a_i \in \mathbb{Z}, $$ for $diag(t^{a_1},t^{a_2},\ldots,t^{a_n}) \in T.$
The algebra of polynomial $T$-invariants $\mathbb{C}[x_1,x_2, \ldots,x_n]^T$ is finitelly generated monomial albera and its minimal generating set consists of elements of the form $x_1^{k_1},x_2^{k_2},\ldots,x_n^{k_n}$ where $(k_1,k_2,\ldots, k_n)$ runs over Hilbert basis of non-negative solutions of the equation $ a_1 k_1+a_2k_2+\cdots+a_n k_n=0.$ See for example the book [Stanley, R. , Combinatorics and commutative algebra].
For example, for $n=3$ let us consider the torus $T=diag(t,t,t^{-2})$. We get the equation $k_1+k_2-2k_3=0$ and its Hilbert basis consists of solutions $(1,1,1),(2,0,1),(0,2,1)$. Thus $$\mathbb{C}[x_1,x_2, x_3]^T=\mathbb{C}[x_1 x_2 x_3,x_1^2x_3,x_2^2x_3].$$
Question 1. What is the algebra of rational invariants $\mathbb{C}(x_1,x_2, x_3)^T$ for $T=diag(t,t,t^{-2})$ as in the example above?
Question 2. What we know about a (minimal) generating set of the algebra of rational torus invariants $\mathbb{C}(x_1,x_2, \ldots,x_n)^T?$
