Definition:Vanishing Ideal of Subset of Affine Space
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Definition
Let $k$ be a field.
Let $n \ge 0$ be a natural number.
Let $k \sqbrk {X_1, \ldots, X_n}$ 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:
- $\map I S = \set {f \in k \sqbrk {X_1, \ldots, X_n} : \forall x \in S : \map f x = 0}$
Also known as
The vanishing ideal of $S$ is also know as its associated ideal.