Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n \in \mathbb N$, and let $X \subset \mathbb A_k^n$ be a nonempty affine algebraic set. Define the vanishing ideal $I(X) \trianglelefteq k[x_1,\ldots,x_n]$ by
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\begin{align*}
I(X)=\{f \in k[x_1,\ldots,x_n] : f(a)=0 \text{ for every } a\in X\}.
\end{align*}
Let $\mathcal O(X)$ denote the ring of regular $k$-valued functions on $X$ in the Zariski sense: a function $\varphi:X\to k$ belongs to $\mathcal O(X)$ if, for every $a\in X$, there exist a Zariski open neighbourhood $U\subset X$ of $a$ and polynomials $p,q\in k[x_1,\ldots,x_n]$ such that $q(b)\ne 0$ for every $b\in U$ and