Definition:Zero Locus of Set of Polynomials
(Redirected from Definition:Zero Locus of Ideal of Polynomials)
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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
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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 $\map Z I$.