Let $f_1,f_2, \ldots ,f_n$ be polynomials in any number of variables with algebraic coefficients. Is there algorithm to determine whether the ring $\mathbb{Z}[f_1,f_2,\ldots ,f_n]$ contains a non-constant polynomial with integer coefficients?

To illustrate what the game is here, I'll give some examples:

$\mathbb{Z}[\sqrt{2}x-y]\cap \mathbb{Z}[x,y]=\mathbb{Z}$. Proof-sketch: Suppose that $H\in\mathbb{Z}[u]$ and $H(\sqrt{2}x-y)\in\mathbb{Z}[x,y]$. Choose a sequence of distinct points $(x_n,y_n)\in\mathbb{Z}^2$ such that $\sqrt{2}x_n-y_n$ converges. Then $H(\sqrt{2}x_n-y_n)$ is eventually an integer constant $c$. Thus the equation $H(u)=c$ has infinitely many solutions. It follows that $H$ is identically $c$

$\mathbb{Z}[\sqrt{2}x-y, 2\sqrt{2}xy-z]$ contains the polynomial $2x^2+y^2-z$. (Take the square of the first polynomial plus the second.) Note that the generators are algebraically independent.

$\mathbb{Z}[\sqrt{2}x-y, \sqrt{3}xy-z]\cap \mathbb{Z}[x,y,z]=\mathbb{Z}$. (I know only a rather lengthy and ad hoc proof)