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

Pezda, T.

Manuscr. Math. 83, No. 3-4, 279-289 (1994).

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 \(N=1\).

Reviewer: Władysław Narkiewicz (Wrocław)

MSC:

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

Keywords:

polynomial maps; periodic points; finite orbits

Citations:

Zbl 0803.11063

References:

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