Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n\in\mathbb N$, and let $R:=k[x_1,\dots,x_n]$. For an ideal $J\trianglelefteq R$, write
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\begin{align*}
V(J):=\{a\in\mathbb A_k^n:f(a)=0\text{ for every }f\in J\}.
\end{align*}
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Let $X\subseteq \mathbb A_k^n$ be an affine algebraic set, let
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\begin{align*}
I(X):=\{f\in R:f(a)=0\text{ for every }a\in X\},
\end{align*}
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and let
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\begin{align*}
k[X]:=R/I(X)
\end{align*}
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be its coordinate ring. For each point $a=(a_1,\dots,a_n)\in X$, define