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It is know that an injective polynomial map $f:\overline{\mathbb{Q}}^{n} \longrightarrow \overline{\mathbb{Q}}^{n}$ is an bijection with inverse regular (Cynk-Rusek theorem). My question is following:

Take $\mathcal{O}$ the integral closure $\mathbb{Z}$ in $\overline{\mathbb{Q}}$. Let $F: \mathcal{O}^{n} \longrightarrow \mathcal{O}^{n}$ an injective polynomial map with $\det J_{f} =1$ (jacobian condition) and coefficients in $\mathbb{Z}$ . Is $F$ an invertible map?

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    $\begingroup$ That is true. Your map is a morphism of schemes, $f:\mathbb{A}^n_{\mathbb{Z}}\to \mathbb{A}^n_{\mathbb{Z}}$. Since the determinant of the Jacobian is invertible, the morphism $f$ is 'etale (not necessarily finite, which is the most common mistake in attempted solutions of the Jacobian conjecture). If the induced map $F$ is injective, then $f$ is an open embedding. If the complement is nonempty, then after reduction to some finite field $\mathbb{F}_{p^r}$, the complement has a rational point. By the Dirichlet box principle, the reduction is non-injective, contradicting that $f$ embeds. $\endgroup$ Commented Dec 1, 2017 at 19:39
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    $\begingroup$ I just realized that injectivity of $F$ is not equivalent to injectivity of $\overline{\mathbb{Q}}^n \to \overline{\mathbb{Q}}^n$. For instance, $F(x) = 4x^2-x$ is injective on $\mathcal{O}$, but it is not injective on $\mathbb{Q}$. So the argument in my previous comment is incomplete. $\endgroup$ Commented Dec 1, 2017 at 20:22
  • $\begingroup$ @JasonStarr: thanks for the argument. In case, $f(x) = 4x^{2}-x$ the jacobian condition ($\det J_{f} = 1$) isn't satisfied. $\endgroup$
    – numberwat
    Commented Dec 1, 2017 at 21:20
  • $\begingroup$ Indeed, the Jacobian condition is not satisfied for $F(x)=4x^2-x$. However, my point is that the condition regarding injectivity on $\mathcal{O}^n$ does not imply the condition regarding injectivity on $\overline{\mathbb{Q}}^n$. $\endgroup$ Commented Dec 1, 2017 at 21:59
  • $\begingroup$ for me it isn't clear that $F$ injective implies that $f$ is open embedding. How to prove it? Is this a general fact? $\endgroup$
    – numberwat
    Commented Dec 7, 2017 at 17:07

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