Fixed Points of Polynomials over Division Rings

TL;DR

This paper investigates fixed points of polynomials over division rings, establishing conditions for their existence and behavior, and extending classical polynomial dynamics to non-commutative algebraic structures.

Contribution

It characterizes fixed points of polynomials over division rings and provides conditions for periodic points, extending polynomial dynamics beyond commutative fields.

Findings

A polynomial of degree m ≥ 2 has at most m conjugacy classes of fixed points.

Fixed points satisfy f(λ)=λ, similar to the commutative case.

Periodic points may behave differently in division rings than in commutative fields.

Abstract

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings $D$. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda)=\lambda$ for any $n \in \mathbb{N}$, where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$. Periodic points are similarly defined. We prove that $\lambda$ is a fixed point of $f(x)$ if and only if $f(\lambda)=\lambda$, which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most $m$ conjugacy classes of fixed points. We also consider arbitrary periodic points, and show that in general, they do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.