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## Acta Arithmetica

1994 | 66 | 1 | 11-22
Tytuł artykułu

### Polynomial cycles in certain local domains

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
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.
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Rocznik
Tom
Numer
Strony
11-22
Opis fizyczny
Daty
wydano
1994
otrzymano
1992-09-25
poprawiono
1993-08-09
Twórcy
autor
• Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
• [1] W. Narkiewicz, Polynomial cycles in algebraic number fields, Colloq. Math. 58 (1989), 151-155.
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