Commuting polynomials and self-similarity
type: Item
instance of: scholarly article
title: Commuting polynomials and self-similarity
zbMATH Open document ID: 1109.12003
author: N. E. Zubov
published in: The New York Journal of Mathematics
publication date: 2007-03-17T00:00:00Z
full work available at URL: https://eudml.org/doc/127085
full work available at URL: 13-5.html
review text: As a corrolary of their studies of functions of a complex variable, between 1922 and 1924, Julia and Ritt obtained a classification of pairs of polynomials with complex coefficients which commute over composition; but their proof used topological and algebraic methods. In this paper the author comes back to this question using only very elementary techniques. Let \(F\) be an algebraically closed field of characteristic zero and \(f \in F[x]\) a polynomial of degree \(>1\). The author shows that the number of polynomials \(P\in F[x]\) of degree \(k\) which commute with \(f\) is either zero or equal to the number of linear polynomials in \(F[x]\) which commute with \(f\). As a corollary, he obtains a theorem of E.A. Bertram characterizing those polynomials commuting with a Chebyshev polynomial.
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 12E10
zbMATH DE Number: 5134557
zbMATH Keywords: polynomial
zbMATH Keywords: field
zbMATH Keywords: Chebyshev polynomial
reviewed by: Maurice Mignotte
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5134557