Fixed Points of Polynomials Over Division Rings

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 show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

Roots and Dynamics of Octonion Polynomials

“…In the second part of the paper, we show that if f (x) is a quadratic monic polynomial over an octonion algebra and f (α) = α, then f •n (α) = α for all n ∈ N (this was shown to be true for all polynomials over associative division algebras but false over octonion algebras in [5]), and we determine when a given fixed point of such a polynomial over O or H is attracting, repelling or neutral. The last part provides more information about pseudo-periodic points, generalizing certain aspects from the theory of fixed points.…”

“…In particular, unlike the commutative case, there is in general no equality between f •n (α) and f * n (α) (see [5]).…”

“…We say that α is a "fixed point" of f (x) if f •n (α) = α for every n ∈ N. It was proven in [5] that if f (α) = α where f is defined over a division algebra, then α is a fixed point. As we demonstrate in Section 4, this is also true for monic quadratic octonion polynomials.…”

“…Some recent attempts have been made to generalize certain aspects of this theory to polynomials over the quaternions, such as [2] and [10]. More recently, the authors of this article attempted to generalize some aspects of this theory to general division rings and octonion algebras (see [5]). The study of octonion polynomials was also suggested in [7] in the context of polynomials over Lie algebras.…”

“…The study of octonion polynomials was also suggested in [7] in the context of polynomials over Lie algebras. It is important to note that the composition of such polynomials is non-associative, and in general f • f (λ) f ( f (λ)) for λ ∈ H, see Section 2 and [5] for details.…”