Polynomials over finite fields which commute with a permutation polynomial
type: Item
instance of: scholarly article
title: Polynomials over finite fields which commute with a permutation polynomial
zbMATH Open document ID: 0799.11057
author: Chong-Yun Chao
published in: Journal of Algebra
publication date: 1994-11-20T00:00:00Z
review text: For \(q\) a prime power, let \(F_ q\) denote the finite field of order \(q\) and let \(F_ q[x]\) be the ring of polynomials in a single indeterminate over \(F_ q\). For any permutation polynomial \(f\in F_ q[x]\), (\(1-1\) mapping of \(F_ q\) onto itself), the author uses properties of the centralizer ring of a permutation group to characterize those polynomials \(g\in F_ q[x]\) with the property that \(f(g(x))= g(f(x))\). Pólya and de Bruijn enumeration techniques are then used to determine the numbers of such polynomials \(g\) and permutation polynomials \(g\in F_ q[x]\) of degree \(<q\). An algorithm is also given to determine all such polynomials \(g\). As the author points out on page 296, since the centralizer ring of a permutation group and Pólya and de Bruijn enumeration techniques apply to finite sets, one can easily consider extensions to much more general settings. The setting of finite fields is considered here because this special case was first considered by \textit{C. Wells} [Proc. Am. Math. Soc. 46, 347-350 (1974; Zbl 0298.12009)] for translations \(f(x)= x+a\), and the reviewer [Proc. Am. Math. Soc. 84, 315-317 (1982; Zbl 0498.12018)] for linear permutations \(f(x)= bx+a\).
reviewed by: Gary L. Mullen
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 05A18
zbMATH DE Number: 554208
zbMATH Keywords: finite field
zbMATH Keywords: polynomials
zbMATH Keywords: permutation polynomial
zbMATH Keywords: permutation group
zbMATH Keywords: enumeration
MaRDI profile type: Publication
full work available at URL: jabr.1994.1019
OpenAlex ID: W2010491695
DOI: 10.1006/JABR.1994.1019
description: scientific article; zbMATH DE number 554208