Let $k$ be a field, let $n \geq 0$ be an integer, and set $R := k[x_0,\ldots,x_n]$. Let $S \subset R$ be a set of homogeneous polynomials, and let $I := (S) \trianglelefteq R$ be the ideal generated by $S$. For a set $T$ of homogeneous polynomials in $R$, define
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\begin{align*}
V_+(T) := \{[a_0:\cdots:a_n] \in \mathbb{P}^n_k : f(a_0,\ldots,a_n)=0 \text{ for every } f \in T\}.
\end{align*}
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For the homogeneous ideal $I$, interpret $V_+(I)$ as the common zero locus of the homogeneous elements of $I$. Then