Zero locus
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Revision 327
7/8/2026, 7:52:19 AM · Ancient Tree
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The zero locus, or vanishing set, of a set $S\subseteq k[x_{1},\ldots,x_{n}]$ [[Polynomial ring|of polynomials]], is defined as the following subset of [[Affine space over a field|affine space]]:2
$$V(S)=\left\{p \in \mathbb{A}_k^n: f(p)=0 \text { for every } f \in S\right\}$$2
$$V(S)=\left\{p \in \mathbb{A}_k^n: f(p)=0 \text { for every } f \in S\right\}.$$3
##### Remarks4
- If $S=\{P\}$, one simply writes $V(S)=V(P)$.Revision 325
7/8/2026, 7:48:36 AM · Ancient Tree
Concept created
The zero locus, or vanishing set, of a set $S\subseteq k[x_{1},\ldots,x_{n}]$ [[Polynomial ring|of polynomials]], is defined as the following subset of [[Affine space over a field|affine space]]:
$$V(S)=\left\{p \in \mathbb{A}_k^n: f(p)=0 \text { for every } f \in S\right\}$$