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 ƒ1,…, ƒm are elements of the free associative algebra D〈 Χ1,…,Χm〉 generated by m ≥ 1 variables over an algebraically closed division ring D of finite dimension over its center F, and if the induced map ƒ = (ƒ1,…ƒm) : Dm → Dm 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〈 Χ1,…,Χm〉 with zero constant term such that p(F) ≠ {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 Mn(D) can be expressed as a difference of pairs of multiplicative commutators of elements from p(Mn(D)), provided again that D is finite-dimensional over F.
Paran, E. and Tran, N.S. (2026) Journal of Pure and Applied Algebra, 230(2), 108186.