Note on the local growth of iterated polynomials
type: Item
instance of: scholarly article
title: Note on the local growth of iterated polynomials
zbMATH Open document ID: 0553.26003
DOI: 10.1007/BF02192665
published in: Aequationes Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137018
review text: Let f be a real polynomial and let \(x_ i\), for \(i=1,2,3,...\), be defined by \(x_{i+1}=f(x_ i),\) where \(x_ i\) is a real number. The authors have proved that if f is of degree n and \(x_{i+1}-x_ i\geq 1\) for \(i=1,...,n+1\) then, for every \(i\in \{1,...,n\}\) and \(k\in \{1,...,n- i+1\},\) \[ -2^{k-1}/k!<f[x_ i,...,x_{i+k}]<(x_{i+k+1}- x_{i+k}+2^{k-1}-1)/k!, \] where \(f[x_ i,...,x_{i+k}]\) denotes the Newton difference quotient for f and \(x_ i,...,x_{i+k}.\)- As a consequence of this theorem the authors have obtained certain bounds on the local increase or decrease of the iteration sequence \((x_ i: i=1,2,3,...).\) Also a number-theoretic result related to a generalization of the van der Waerden numbers has been given as another consequence of the quoted theorem.
reviewed by: Witold Jarczyk
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 26A18
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 3880969
zbMATH Keywords: iterates of polynomials
zbMATH Keywords: Newton difference quotient
zbMATH Keywords: van der Waerden numbers
author: John W. Layman
author: Bruce M. Landman
MaRDI profile type: Publication
cites work: Q3278407
cites work: Generalized van der Waerden numbers
OpenAlex ID: W1989975184
description: scientific article; zbMATH DE number 3880969