I am learning algebraic geometry and have a rather vague question about the the correspondence between polynomial ideals and closed sets in the Zariski-topology in the context of composition of polynomials.
Let $\mathbb{K}$ be an algebraically closed field, polynomials $f,g_1,...,g_n\in\mathbb{K}[x_1,...,x_n]$ and $\overline{\tau}_{\mathbb{A}_\mathbb{K}^n}$ be the set of (Zariski-)closed sets (e.g. $Z(f)\in\overline{\tau}_{\mathbb{A}_\mathbb{K}^n}$ is the zero locus or algebraic set of the ideal $(f)$...). As we have basic identities like $Z(g_1\cdot g_2)=Z(g_1)\cup Z(g_2)$ and $Z(g_1,g_2)=Z(g_1)\cap Z(g_2)$ I wonder if there are also identities of the kind \begin{align} Z(f(g_1,...,g_n))=\theta(Z(f),Z(g_1),...,Z(g_n)) \end{align} for some $n+1$-ary function $\theta:\left(\overline{\tau}_{\mathbb{A}_\mathbb{K}^n}\right)^{n+1}\to\overline{\tau}_{\mathbb{A}_\mathbb{K}^n}$. I did never encounter this situation in my algebraic geometry literature. Is there any material on this?
$g_1, \cdots, g_n$ together define a morphism $g: \Bbb K^n \to \Bbb K^n$.
$$\begin{array}{cl} & Z(f(g_1, \cdots, g_n)) \\ =& \{ x \in \Bbb K^n \mid f(g_1(x), \cdots, g_n(x)) = 0 \} \\ =& \{ x \in \Bbb K^n \mid (g_1(x), \cdots, g_n(x)) \in Z(f) \} \\ =& \{ x \in \Bbb K^n \mid g(x) \in Z(f) \} \\ =& g^{-1}(Z(f)) \end{array}$$
I am afraid there might be no expression strictly of the form you want; this is the best I can do.