Definition:Zero Locus of Set of Polynomials

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 $\map Z I$.

Also see

 * Definition:Zero Locus of Subset of Ring
 * Definition:Vanishing Ideal of Subset of Affine Space
 * Definition:Affine Algebraic Set
 * Definition:Affine Algebraic Variety