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

TL;DR

This paper extends the Ax-Grothendieck theorem to polynomial maps over algebraically closed division rings, establishing surjectivity from injectivity and exploring polynomial images and Waring type decompositions.

Contribution

It generalizes the classical Ax-Grothendieck theorem to division rings and provides new results on polynomial images and matrix decompositions over such rings.

Findings

Injective polynomial maps are surjective over algebraically closed division rings of finite dimension.

Certain polynomial images over division rings are dense, covering the entire ring.

Matrices over division rings can be expressed as differences of commutators within polynomial images.

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 $.