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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 $\mathbb R$.

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  • $\begingroup$ With $z,x_j$ where (in $\mathbb Q,\mathbb R,\dots$)? $\endgroup$ – fedja Jun 12 at 17:57
  • $\begingroup$ all of them should be in $\mathbb Q$ $\endgroup$ – afiori Jun 12 at 21:04
  • $\begingroup$ Getting $Z_P = \mathbb{Z}$ is a well-known open problem (existential definition of $\mathbb{Z}$ in $\mathbb{Q}$) and possibly $Z_P = \mathbb{N}$ is no easier, but I am not an expert. See the paper of Koenigsmann in the Annals 2016 for the latest partial results on this. $\endgroup$ – Felipe Voloch Jun 13 at 1:55

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