Primitive prime divisors in zero orbits of polynomials
type: Item
instance of: scholarly article
zbMATH Open document ID: 1273.37045
author: Kevin Doerksen
author: Anna Haensch
published in: Integers
publication date: 2012-07-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.3971
full work available at URL: http://citeseerx.ist.psu.edu/viewdoc/summary
full work available at URL: vol12.html
Mathematics Subject Classification ID: 37P05
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 6054420
zbMATH Keywords: arithmetic dynamics
zbMATH Keywords: polynomial maps
zbMATH Keywords: orbits
zbMATH Keywords: primitive prime divisors
zbMATH Keywords: rigid divisibility sequences
MaRDI profile type: Publication
OpenAlex ID: W2963540269
arXiv ID: 1009.3971
DOI: 10.1515/INTEGERS-2011-0117
title: Primitive prime divisors in zero orbits of polynomials
review text: The authors study primitive prime divisors in the orbit of zero under iterated application of a unicritical polynomial \(\varphi(z) = z^d+c\), with \(c,d \in \mathbb{Z}\). The main theorem states that, when \(d \geq 2\) and \(0\) is a wandering point of \(\varphi\), then \(\varphi^n(0)\) has a primitive prime divisor for all \(n \geq 1\), except of course for \(n = 1\) when \(c = \pm 1\). (Zero is not wandering exactly when \(c=0\), or \(c=-1\) and \(d\) is even, or \(c=-2\) and \(d=2\).) Here, a primitive prime divisor is a prime that divides \(\varphi^n(0)\) but not \(\varphi^k(0)\) for any \(k < n\). This improves a theorem of \textit{B. Rice} [Integers 7, No. 1, A26, 16 p. (2007; Zbl 1165.11028)]. The proof is based on Rice's observation that these sequences are rigid divisibility sequences.
reviewed by: Wolfgang Steiner
description: scientific article; zbMATH DE number 6054420