Combinatorial cycles of a polynomial map over a commutative field
type: Item
instance of: scholarly article
title: Combinatorial cycles of a polynomial map over a commutative field
zbMATH Open document ID: 0596.12020
DOI: 10.1016/0012-365X(86)90024-5
author: Guy Chassé
published in: Discrete Mathematics
publication date: 1986-01-01T00:00:00Z
review text: A map g from a field K into itself is said to have a cycle of length n if there exist n distinct elements \(x_ 1,...,x_ n\) of K such that \(g(x_ i)=x_{i+1}\) for \(1\leq i\leq n-1\) and \(g(x_ n)=x_ 1\). Let K be algebraically closed and let \(f\in K[x]\) with deg(f)\(\geq 2\). Let t be a prime number different from the characteristic of K and different from the multiplicative orders of the roots of unity u of the form \(u=f'(c)\) with \(f(c)=c\). Then it is shown that the map from K into itself induced by f has a cycle of length t. For finite fields K the action of the Frobenius automorphism on the cycles of polynomial maps is also studied.
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 05A99
zbMATH DE Number: 3959581
zbMATH Keywords: cycle length
zbMATH Keywords: finite fields
zbMATH Keywords: Frobenius automorphism
zbMATH Keywords: cycles of polynomial maps
MaRDI profile type: Publication
cites work: Q3935355
cites work: A monte carlo method for factorization
full work available at URL: 0012-365x(86)90024-5
OpenAlex ID: W2113644636
description: scientific article; zbMATH DE number 3959581