Regular Functions What is a general technique for finding the ring of functions on an open subset of affine space over an algebraically closed field? I.e. "Find the ring of regular functions on $ \mathbb{A}_k^n- \{(0,...,0)\}$." Or find the ring of regular functions on $U_f$ a distinguished open subset? 
 A: I think that there is no general method. But the following two statements are fundamental and useful. I will state them in the scheme language (of course they hold, in particular, for varieties):
1) If $X=\mathrm{Spec}(A)$ is an affine scheme and $f \in A$, then $X_f$ is isomorphic to the affine scheme $\mathrm{Spec}(A_f)$. This also explains the notation $X_f$. In particular, the ring of regular functions on $X_f$ is $A_f$.
2) (Algebraic Hartogs Lemma) If $X$ is a noetherian normal scheme and $U \subseteq X$ is an open subset such that $X \setminus U \hookrightarrow X$ codimension at least $2$, then the restriction morphism $\Gamma(X,\mathcal{O}_X) \to \Gamma(U,\mathcal{O}_X)$ is an isomorphism.
In particular, for $n \geq 2$, the ring of regular functions on $\mathbb{A}^n \setminus \{0\}$ is the same as for $\mathbb{A}^n$, i.e. $k[x_1,\dotsc,x_n]$. A direct computation is also possible, use the affine cover given by the basic open subsets $\mathbb{A}^n_{x_i}$ and observe $\cap_i k[x_1,\dotsc,x_n]_{x_i} = k[x_1,\dotsc,x_n]$.
