my question can be seen as a extension to this question.

Let $\overline{\mathbb{Q}}$ denote an algebraic closure of $\mathbb{Q}$. Given a polynomial with algebraic coefficients $f \in \overline{\mathbb{Q}}[X_1,...,X_n]$, say $$f=\sum_{(\alpha_1,...,\alpha_n)}c_{(\alpha_1,...,\alpha_n)}X^{\alpha_1}\cdot...\cdot X^{\alpha_n}.$$
Let $(b_1,...,b_n)$ be a root of this polynomial, where $b_1,...b_n \in \mathbb{C}$ (so the $b_{i}$ are **not necessarily algebraic numbers individually**).
By root i mean that we have $$f(b_1,...,b_n) = \sum_{(\alpha_1,...,\alpha_n)}c_{(\alpha_1,...,\alpha_n)}b^{\alpha_1}\cdot...\cdot b^{\alpha_n} = 0.$$
Is it possible to construct a polynomial with rational coefficients $g \in {\mathbb{Q}}[X_1,...,X_n]$, for which $(b_1,...,b_n)$ is a root?

I would already be happy with hints to relevant literature. Thank you!