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

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 f₁,…,f_m are elements of the free associative algebra D〈X₁,…,X_m〉 generated by m≥1 variables over an algebraically closed division ring D of finite dimension over its center F , and if the induced map f=(f₁,…,f_m):D^m→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〈X₁,…,X_m〉 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 M_n(D) can be expressed as a difference of pairs of multiplicative commutators of elements from p(M_n(D)), provided again that D is finite-dimensional over F .