Cycle-lengths of a class of monic binomials
type: Item
instance of: scholarly article
title: Cycle-lengths of a class of monic binomials
zbMATH Open document ID: 1262.11090
author: Władysław Narkiewicz
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2010-07-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1277811639
review text: This paper is a contribution to research on the Morton-Silverman uniform boundedness conjecture in arithmetic dynamics. This conjecture states that the number of pre-periodic points of a rational map in an algebraic number field is bounded by a constant depending only on the dimension of the space, the degree of the map, and the degree of the field. Very little in known about this conjecture, even in one dimension; weaker conjectures have been formulated on bounds for the period of periodic orbits. The author considers the one-dimensional map \(f(x)=x^n+c\) over an algebraic number field \(K\) of degree \(N\), where the degree \(n\) of the map is the power of a prime \(p\). He proves that if \(K\) does not contain \(p\)-th primitive roots of unity, then the period of the periodic orbits of \(f\) in \(K\) is bounded by a constant depending only on \(K\) and \(p\), and if \(p>2^N\), this constant depends only on \(N\).
reviewed by: Franco Vivaldi
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 37P15
Mathematics Subject Classification ID: 37P45
zbMATH DE Number: 5758463
zbMATH Keywords: polynomial cycles
zbMATH Keywords: algebraic number fields
zbMATH Keywords: Morton-Silverman conjecture
MaRDI profile type: Publication
OpenAlex ID: W2070649897
cites work: An Elementary Product Identity in Polynomial Dynamics
cites work: Canonical heights on projective space
cites work: PERIODIC ORBITS OF QUADRATIC POLYNOMIALS
cites work: Cycles of quadratic polynomials and rational points on a genus-2 curve
cites work: Arithmetic properties of periodic points of quadratic maps
cites work: Q4697833
cites work: Q4821147
cites work: Polynomial cycles in certain local domains
cites work: Rational Periodic Points of the Quadratic Function Q c (x) = x 2 + c
cites work: Rational 6-Cycles Under Iteration of Quadratic Polynomials
DOI: 10.7169/FACM/1277811639
Recommended article: On a class of monic binomials
Recommended article: Q3710669
Recommended article: On the number of sets of cycle lengths
Recommended article: Q3306248
Recommended article: ON THE MONOGENITY OF CERTAIN BINOMIAL COMPOSITIONS
Recommended article: Q4268147
Recommended article: On cyclic 〈m,n〉-enumerations
Recommended article: On binary cyclotomic polynomials
Recommended article: The cycle structure of a class of permutation polynomials
Recommended article: Q3474736
description: scientific article; zbMATH DE number 5758463