A local-global principle for surjective polynomial maps

Abstract

Let $R$ be an affine domain of characteristic zero with finite quotients. We prove that a polynomial map over $R$ is surjective if and only if it is surjective over $\hat{R_{\mathfrak{m}}}$, the completion of $R$ with respect to $\mathfrak{m}$, for every maximal ideal $\mathfrak{m} \subseteq R$. In fact, the completions $\hat{R_{\mathfrak{m}}}$ may be replaced by arbitrary subrings containing $R$. We use this result to yield a characterization of surjective polynomial maps, and remark that there does not exist a similar principle for injective polynomial maps.