Polynomial cycles in certain local domains
Volume 66 / 1994
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.
