Commuting polynomials and self-similarity
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.
scientific article; zbMATH DE number 5134557