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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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