# Polynomial cycles in certain local domains

Acta Arithmetica (1994)

• Volume: 66, Issue: 1, page 11-22
• ISSN: 0065-1036

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## Abstract

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1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple $x₀,x₁,...,{x}_{k-1}$ of distinct elements of R is called a cycle of f if $f\left({x}_{i}\right)={x}_{i+1}$ for i=0,1,...,k-2 and $f\left({x}_{k-1}\right)=x₀$. The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number ${7}^{7·{2}^{N}}$, depending only on the degree N of K. In this note we consider the case when R is a discrete valuation domain of zero characteristic with finite residue field. We shall obtain an upper bound for the possible lengths of cycles in R and in the particular case R=ℤₚ (the ring of p-adic integers) we describe all possible cycle lengths. As a corollary we get an upper bound for cycle lengths in the ring of integers in an algebraic number field, which improves the bound given in [1]. The author is grateful to the referee for his suggestions, which essentially simplified the proof in Subsection 6 and improved the bound for C(p) in Theorem 1 in the case p = 2,3.

## How to cite

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T. Pezda. "Polynomial cycles in certain local domains." Acta Arithmetica 66.1 (1994): 11-22. <http://eudml.org/doc/206588>.

@article{T1994,
abstract = {1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple $x₀,x₁,...,x_\{k-1\}$ of distinct elements of R is called a cycle of f if $f(x_i) = x_\{i+1\}$ for i=0,1,...,k-2 and $f(x_\{k-1\}) = x₀$. The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number $7^\{7·2^N\}$, depending only on the degree N of K. In this note we consider the case when R is a discrete valuation domain of zero characteristic with finite residue field. We shall obtain an upper bound for the possible lengths of cycles in R and in the particular case R=ℤₚ (the ring of p-adic integers) we describe all possible cycle lengths. As a corollary we get an upper bound for cycle lengths in the ring of integers in an algebraic number field, which improves the bound given in [1]. The author is grateful to the referee for his suggestions, which essentially simplified the proof in Subsection 6 and improved the bound for C(p) in Theorem 1 in the case p = 2,3.},
author = {T. Pezda},
journal = {Acta Arithmetica},
keywords = {polynomial maps; polynomial cycles; discrete valuation domain; cycle-lengths},
language = {eng},
number = {1},
pages = {11-22},
title = {Polynomial cycles in certain local domains},
url = {http://eudml.org/doc/206588},
volume = {66},
year = {1994},
}

TY - JOUR
AU - T. Pezda
TI - Polynomial cycles in certain local domains
JO - Acta Arithmetica
PY - 1994
VL - 66
IS - 1
SP - 11
EP - 22
AB - 1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple $x₀,x₁,...,x_{k-1}$ of distinct elements of R is called a cycle of f if $f(x_i) = x_{i+1}$ for i=0,1,...,k-2 and $f(x_{k-1}) = x₀$. The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number $7^{7·2^N}$, depending only on the degree N of K. In this note we consider the case when R is a discrete valuation domain of zero characteristic with finite residue field. We shall obtain an upper bound for the possible lengths of cycles in R and in the particular case R=ℤₚ (the ring of p-adic integers) we describe all possible cycle lengths. As a corollary we get an upper bound for cycle lengths in the ring of integers in an algebraic number field, which improves the bound given in [1]. The author is grateful to the referee for his suggestions, which essentially simplified the proof in Subsection 6 and improved the bound for C(p) in Theorem 1 in the case p = 2,3.
LA - eng
KW - polynomial maps; polynomial cycles; discrete valuation domain; cycle-lengths
UR - http://eudml.org/doc/206588
ER -

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