# Multivariate polynomial with infinite but discrete roots on one variable

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 $$\mathbb R$$.

• With $z,x_j$ where (in $\mathbb Q,\mathbb R,\dots$)? – fedja Jun 12 at 17:57
• all of them should be in $\mathbb Q$ – afiori Jun 12 at 21:04
• 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. – Felipe Voloch Jun 13 at 1:55