definition. affine variety [cox1997ideals, 1.2.1] [ag-0006]

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Let \(f_1, \ldots , f_s\) be polynomials in \(K\left [x_1, \ldots , x_n\right ]\). The affine variety defined by \(f_1, \ldots , f_s\) is \[ \mathbf {V}\left (f_1, \ldots , f_s\right )=\left \{\left (a_1, \ldots , a_n\right ) \in \mathbb {A}_K^n \mid f_i\left (a_1, \ldots , a_n\right )=0 \text { for all } 1 \leq i \leq s\right \} \]

Obviously, it is the set of all solutions of the system of polynomial equations \(f_1\left (x_1, \ldots , x_n\right )=\cdots =f_s\left (x_1, \ldots , x_n\right )=0\).