Definition:Zero Locus of Set of Polynomials

Definition
Let $k$ be a field.

Let $n\geq1$ be a natural number.

Let $A = k \left[{X_1, \ldots, X_n}\right]$ 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:


 * $V \left({I}\right) = \left\{{x \in k^n : \forall f \in I: f \left({x}\right) = 0}\right\}$

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 $Z(I)$.

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