Cycles of polynomial mappings in several variables. (English) Zbl 0804.11059

Let \(R\) be a discrete valuation domain of zero characteristic and with finite residue field. It is shown that there exists an upper bound for the length of a finite cyclic orbit of any mapping \(R^ N \to R^ N\) defined by a system of polynomials over \(R\). This bound is given explicitly and depends only on \(R\) and \(N\). This implies a corresponding bound in the case when \(R\) is the ring of integers of an algebraic number field \(K\). Here the bound depends only on \(N\) and the degree of \(K\). This generalizes previous results of the author [Acta Arith. 66, 11–22 (1994; Zbl 0803.11063)] dealing with the case \(N=1\).


37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P20 Dynamical systems over non-Archimedean local ground fields
11S05 Polynomials
37P35 Arithmetic properties of periodic points
13F30 Valuation rings
14E05 Rational and birational maps


Zbl 0803.11063
Full Text: DOI EuDML


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