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FIXED POINTS OF POLYNOMIALS OVER DIVISION RINGS

Published online by Cambridge University Press:  01 March 2021

ADAM CHAPMAN
Affiliation:
School of Computer Science, Academic College of Tel-Aviv-Yaffo, Rabenu Yeruham St., PO Box 8401, Yaffo6818211, Israel e-mail: adam1chapman@yahoo.com
SOLOMON VISHKAUTSAN*
Affiliation:
Department of Computer Science, Tel-Hai Academic College, Upper Galilee, Qiryat Shemona1220800, Israel

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.

Type
Research Article
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author acknowledges the receipt of the Chateaubriand Fellowship (969845L) offered by the French Embassy in Israel.

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