Let $k$ be a field, let $n \in \mathbb{N}$, set $R := k[x_1, \ldots, x_n]$, and let $\mathbb{A}^n_k$ denote affine $n$-space over $k$. For every subset $T \subset R$, define its vanishing set by
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\begin{align*}
V(T) := \{a \in \mathbb{A}^n_k : f(a)=0 \text{ for every } f \in T\}.
\end{align*}
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For every subset $S \subset R$, if $(S) \trianglelefteq R$ denotes the ideal generated by $S$, then