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

TL;DR

This work extends the Ax-Grothendieck theorem to noncommutative polynomial maps over algebraically closed division rings. By reducing to real-closed-field settings via Baer’s structure theory and Wilczyński’s framework, it proves that an injective noncommutative polynomial map $f=(f_1, frac{} frac f_m)$ with $f_i ext{ in }Drangle X_1, frac{} frac X_mrangle$ is automatically surjective when $D$ is centrally-finite over its center $F$, and that certain central-variable polynomials satisfy $p(D)=D$. The paper also analyzes matrix evaluations: $p( ext{M}_n(D))$ contains all diagonalizable matrices, and every matrix in $ ext{M}_n(D)$ can be written as a product of two elements from $p( ext{M}_n(D))$, with stronger commutator-decomposition results in the finite-dimensional case. Collectively, these results connect algebraic-closure notions, noncommutative polynomials, and matrix decompositions, contributing to Waring-type questions and noncommutative algebraic geometry over 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 $.

Theorems & Definitions (19)

• Proposition 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3: Ax-Grothendieck theorem over division rings
• proof
• Theorem 3.1
• proof
• Corollary 4.1
• Theorem 4.2