I want to know if there exists a polynomial $ P(z, x_1,x_2,\ldots,x_n)$ over the rationals such that the set $$ Z_P = \{z | \exists x_1,\ldots,x_n. P(z, x_1,x_2,\ldots,x_n) = 0 \} \subsetneq \mathbb Q $$ is infinite but with no accumulation points.

Ideally I would like $Z_P = \mathbb N$ or $Z_P = \mathbb Z$, but a infinite discrete subset of the rationals is enough.

The context is to encode nonlinear integer arithmetic in nonlinear rational arithmetic, which is also the reason I need $\mathbb Q$ instead of an otherwise more reasonable $\mathbb R$.