Definition:Vanishing Ideal of Subset of Affine Space

Definition
Let $k$ be a field.

Let $n \geq 0$ be a natural number.

Let $k[X_1, \ldots, X_n]$ be the polynomial ring in $n$ variables over $k$.

Let $S \subseteq \mathbb A^n_k$ be a subset of the standard affine space over $k$.

Its vanishing ideal is the ideal:
 * $I(S) = \{f \in k[X_1, \ldots, X_n] : \forall x \in S : f(x) = 0\}$

Also known as
The vanishing ideal of $S$ is also know as its associated ideal.

Also see

 * Definition:Zero Locus of Set of Polynomials
 * Hilbert's Nullstellensatz