# Polynomial cycles in certain local domains

Acta Arithmetica (1994)

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

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topT. 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 -

## References

top## Citations in EuDML Documents

top- Tadeusz Pezda, On cycles and orbits of polynomial mappings ${\mathbb{Z}}^{2}\mapsto {\mathbb{Z}}^{2}$
- Zuzana Divišová, On cycles of polynomials with integral rational coefficients
- Robert Benedetto, Jean-Yves Briend, Hervé Perdry, Dynamique des polynômes quadratiques sur les corps locaux
- Joseph H. Silverman, The field of definition for dynamical systems on ${\mathbb{P}}^{1}$
- Władysław Narkiewicz, Polynomial cycles in certain rings of rationals

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