# Definition:Zero Locus of Set of Polynomials

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## Contents

## Definition

Let $k$ be a field.

Let $n\geq1$ be a natural number.

Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the polynomial ring in $n$ variables over $k$.

Let $I \subseteq A$ be a set.

Then the **zero locus** of $I$ is the set:

- $\map V I = \set {x \in k^n : \forall f \in I: \map f x = 0}$

## Remark

Note that this definition applies in particular to the case where $I$ is an ideal. See also Zero Locus of Set is Zero Locus of Generated Ideal.

## Also denoted as

The **zero locus** of $I$ can also be denoted by $\map Z I$.