Let $L$ be a field and $K$ be a subfield of $L.$ Let $V$ be an affine algebraic $K$-variety in $\Bbb A^n (L).$ Consider the vanishing ideal of $V$
$$\begin{align} \mathcal {I} (V) & = \left \{f \in K[X_2,X_2, \cdots , X_n] : f(x_1,x_2, \cdots , x_n) = 0\ \text {for all}\ (x_1,x_2, \cdots , x_n) \in V \right \}. \end{align}$$
Now let us suppose that $V = \varnothing.$ Then what can we say about $\mathcal {I} (V)$? Is it the whole of $K[X_1,X_2, \cdots ,X_n]$ or some other else?
Any help regarding this will be highly appreciated. Thank you very much.