Images of polynomial maps and the Ax-Grothendieck theorem over algebraically closed division rings

Abstract

We study the images of polynomial maps over algebraically closed division rings. Our first result generalizes the classical Ax-Grothendieck theorem: We show that if $ f_1, \ldots, f_m $ are elements of the free associative algebra $ D\langle X_1, \ldots, X_m \rangle $ generated by $ m \geq 1 $ variables over an algebraically closed division ring $ D $ of finite dimension over its center $ F $, and if the induced map $ f = (f_1, \ldots, f_m) \colon D^m \to D^m $ is injective, then $ f $ must be surjective. With no condition on the dimension over the center, our second result is that $ p(D) = D $ if $ p $ is either an element in $ F\langle X_1, \ldots, X_m \rangle $ with zero constant term such that $ p(F) \neq \{0\} $, or a nonconstant polynomial in $F[x]$. Furthermore, we also establish some Waring type results. For instance, for any integer $ n > 1 $, we prove that every matrix in $ \mathrm{M}_n(D) $ can be expressed as a difference of pairs of multiplicative commutators of elements from $ p(\mathrm{M}_n(D)) $, provided again that $ D $ is finite-dimensional over $ F $.