Zero locus

The zero locus, or vanishing set, of a set $S\subseteq k[x_{1},\ldots,x_{n}]$ of polynomials, is defined as the following subset of affine space:
$$V(S)=\left\{p \in \mathbb{A}_k^n: f(p)=0 \text { for every } f \in S\right\}.$$

Remarks

• If $S=\{P\}$, one simply writes $V(S)=V(P)$.