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?