definition. affine variety [gathmann2013commutative, 0.3] [ag-000D]

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Let \(S \subset K\left [x_1, \ldots , x_n\right ]\) be a set of polynomials. The zero locus (or zero set) of \(S\) is \[ V(S):=\left \{x \in \mathbb {A}_K^n: f(x)=0 \text { for all } f \in S\right \} \subset \mathbb {A}_K^n \]

An affine algebraic variety over \(K\) is a subset of \(\mathbb {A}_K^n\) of this form. It's usually simply called an affine variety over \(K\), or an affine \(K\)-variety.

If \(S=\left (f_1, \ldots , f_k\right )\) is a finite set, \(V(S)\) can be written as \(V\left (f_1, \ldots , f_k\right )\).

Obviously, it is the set of all solutions of the system of polynomial equations \(f_1\left (x_1, \ldots , x_n\right )=\cdots =f_s\left (x_1, \ldots , x_n\right )=0\) [cox1997ideals, 1.2.1], denoted \(\operatorname {Sol}(S;K)\) [dolgachev2013introduction, p. 1].