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 $f_1,\dots ,f_m$ are elements of the free associative algebra $D⟨X_1,\dots ,X_m⟩$ generated by $m\ge 1$ variables over an algebraically closed division ring $D$ of finite dimension over its center $F$, and if the induced map $f=(f_1,\dots ,f_m):D^m\to D^m$ 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 $F⟨X_1,\dots ,X_m⟩$ with zero constant term such that $p(F)\ne \{0\}$, 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 $F$.