Bijective polynomials $f\in\mathbb Q[X_1,\dots,X_n]$

Question

Does it exist a polynomial $f\in\mathbb Q[X_1,\dots,X_n]$, of degree greater than $1$, that defines a one-to-one correspondence $f:\mathbb Z^n\to\mathbb Z$ or $f:\mathbb N^n\to\mathbb N$?

Are there methods to approach such questions at all?

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See also Conjecture about polynomials $f_n\in\mathbb Q[X_1,\dots,X_n]$ defining bijections $\mathbb N^n\to\mathbb N$

Answer 1

It is known that the polynomial $f(n,m)=\frac{1}{2}(n+m)(n+m+1)+m$ defines bijection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$.

Reference: Polynomial bijections.