---
type: "concept"
title: "Zero locus"
slug: "zero-locus"
language: "en"
translationGroupId: "cmrbs0kha0007pa01cqpcyq76"
domain: "Algebra"
status: "stub"
aliases: ["Vanishing set"]
lastEditedBy: "ancient-tree"
---

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]]:
$$V(S)=\left\{p \in \mathbb{A}_k^n: f(p)=0 \text { for every } f \in S\right\}.$$
##### Remarks
- If $S=\{P\}$, one simply writes $V(S)=V(P)$.