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Polynomial cycles in certain local domains. (English) Zbl 0803.11063
If \(R\) is a domain and \(f\in R[X]\) then a sequence \(a_ 1,\dots, a_ n\) of distinct elements of \(R\) is called an \(f\)-cycle provided \(f(a_ i)= a_{i+1}\) \((i=1,2, \dots, n-1\)) and \(f(a_ n)= a_ 1\). The number \(n\) is called the length of the cycle. The author shows that if \(R\) is a discrete valuation domain of zero characteristic with finite residue class field then the length of \(f\)-cycles for all \(f\in R[X]\) is uniformly bounded. In particular, if \(\mathbb{Z}_ p\) denotes the ring of \(p\)-adic integers then it is shown that a number \(n\) is the length of an \(f\)-cycle for a suitable polynomial over \(\mathbb{Z}_ p\) if and only if \(n\) can be written in the form \(n=ab\), where \(a\) divides \(p-1\) and \(1\leq n\leq p\), with the exception of the cases \(p= 2,3\) when also cycles of length \(p^ 2\) exist. This leads to an essential improvement of the bounds for cycle-lengths in rings of integers of algebraic number fields, obtained by the reviewer [Colloq. Math. 58, 151–155 (1989; Zbl 0703.12002)].
In a later paper [Manuscr. Math. 83, No. 3-4, 279–289 (1994; Zbl 0804.11059)] the author extends his result to mappings \(R^ N\to R^ N\) \((N\geq 2)\) defined by a set of \(N\) polynomials in \(N\) variables.

MSC:
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P20 Dynamical systems over non-Archimedean local ground fields
11S05 Polynomials
13B25 Polynomials over commutative rings
13F30 Valuation rings
13G05 Integral domains
14E05 Rational and birational maps
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