Algebraic dynamics of polynomial maps on the algebraic closure of a finite field. I
type: Item
instance of: scholarly article
title: Algebraic dynamics of polynomial maps on the algebraic closure of a finite field. I
zbMATH Open document ID: 0810.11071
author: Anjula Batra
author: Patrick Morton
published in: Rocky Mountain Journal of Mathematics
publication date: 1994-09-28T00:00:00Z
review text: If \(\sigma\) is a polynomial over a finite field \(F\), then it induces a map \(\widehat{\sigma}\) on the set of all irreducible monic polynomials over \(F\) by putting \(\widehat {\sigma}(g)=f\) if for every root \(\alpha\) of \(g\) in the algebraic closure \(\widehat{F}\) of \(D\) one has \(f(\sigma (\alpha))=0\). The authors prove that if \(\sigma\) is non-constant, then \(\widehat {\sigma}(x)\) has infinitely many fixed-points. Then the case \(\sigma(x)= x^ q+ ax\) (where \(q\) is the cardinality of \(F\) and \(a\neq 0\)) is considered and periodic points of the map \(\sigma: \widehat{F}\to \widehat{F}\) are studied, using the polynomial \(\Phi_{m\sigma} (x)= \prod_{d\mid m} (\sigma^ d (x)-x)^{\mu (m/d)}\), (with \(\sigma^ d\) denoting the \(d\)-th iterate of \(\sigma\), which has already shown its utility in such questions in earlier papers of the second author. In particular a formula for the number of periodic points of given order and degree (over \(F\)) is given and an interesting connection with Mersenne primes is established. Remarks: In a recent paper [Manuscr. Math. 83, 279-289 (1994; Zbl 0804.11059)] \textit{T. Pezda} determined the orders of periodic points, up to finitely many, of all polynomials over an algebraically closed field of positive characteristic. [Part II, cf. ibid. 24, No. 3, 905-932 (1994)].
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 12E20
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 645213
zbMATH Keywords: polynomial maps
zbMATH Keywords: fixed points
zbMATH Keywords: polynomial cycles
zbMATH Keywords: dynamical systems
zbMATH Keywords: finite field
zbMATH Keywords: Mersenne primes
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072411
OpenAlex ID: W4233146631
cites work: Q4002072
cites work: Q5620640
cites work: Single variable Bell polynomials
cites work: A Class of Polynomials
cites work: Continued Fractions and Chaos
cites work: Chaotic Bursts in Nonlinear Dynamical Systems
cites work: The Factorization of the Cyclotomic Polynomials mod p
cites work: Explicit Class Field Theory for Rational Function Fields
cites work: Q4694260
cites work: Q3216750
cites work: The Galois Theory of Periodic Points of Polynomial Maps
cites work: Q4001553
cites work: Geometry of linear maps over finite fields
cites work: Dynamics over irreducible polynomials
cites work: Galois theory of periodic orbits of rational maps
DOI: 10.1216/RMJM/1181072411
Recommended article: The Galois Theory of Periodic Points of Polynomial Maps
Recommended article: Periods of maps on irreducible polynomials over finite fields
Recommended article: The dynamics of linearized polynomials
Recommended article: Bifurcations and discriminants for polynomial maps
Recommended article: Q4510780
Recommended article: Q4435135
Recommended article: An Elementary Product Identity in Polynomial Dynamics
Recommended article: Corrigendum: ‘On certain algebraic curves related to polynomial maps, Compositio Math. 103 (1996), 319–350’
Recommended article: Roots and coefficients of polynomials over finite fields
Recommended article: Polynomial cycles in algebraic number fields
description: scientific article; zbMATH DE number 645213