vanishing locus

Given a set $S\subset \mathbb{C}[x_1,\dots ,x_n]$, the vanishing locus of the set $S$, is

$$V(S):=\{z\in {\mathbb{C}}^n|f(z)=0\forall f\in S\}$$

Note, we don’t need to do this where the base field is $\mathbb{C}$. We can find the vanishing locus for any family of polynomials $S$ in the polynomial ring $k[x_1,\dots ,x_n]$ for any algebraically closed field.

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References