Suppose that $f(x_1,\dots,x_n)$ is a polynomial in variables $x_1,\dots, x_n$. Does there always exist an injective homomorphism $$\phi:\mathbb{Z}[x_1,\dots,x_n]/(f)\to\mathbb{Z}[y_1,\dots,y_m]$$ for some $m$?
If not, are there sufficient conditions one can put on $f$ to ensure that $\phi$ exists?
If $\phi$ exists, is there an algorithm to construct it?
Also, I'm not sure if it matters at all, but it is not incredibly important that the coefficient ring of my polynomial rings be $\mathbb{Z}$. If this question has an easier answer working over a different ring, then I'd be happy to hear it as well.
Edit: As several people pointed out, the answer to the first question is of course no. I have a specific example in mind, maybe I should have just asked it initially. Let $$f(x_1,x_2,x_3,x_4,x_5) = x_1 x_3 x_5 + x_2^2 - x_1^2 x_5 -x_2 x_4.$$
Is there an injective map $$\phi:\mathbb{Z}[x_1,x_2,x_3,x_4,x_5]/(f) \to \mathbb{Z}[y_1,\dots,y_m]$$ for some m?