A local-global principle for surjective polynomial maps

TL;DR

This paper establishes a local-global principle for surjective polynomial maps over affine domains of characteristic zero, showing surjectivity can be checked on completions or certain subrings, but not for injectivity.

Contribution

It proves a new criterion linking global surjectivity of polynomial maps to their surjectivity over local completions and subrings, expanding understanding of polynomial map properties.

Findings

Surjectivity over an affine domain is equivalent to surjectivity over all its local completions.

The principle applies to arbitrary subrings containing the domain, not just completions.

No similar principle exists for injective 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.