Polynomial cycles in finitely generated domains
type: Item
instance of: scholarly article
title: Polynomial cycles in finitely generated domains
zbMATH Open document ID: 0840.13003
DOI: 10.1007/BF01293588
author: Franz Halter-Koch
author: Władysław Narkiewicz
published in: Monatshefte für Mathematik
publication date: 1996-07-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178686
review text: For a non-empty set \(\Omega\) and a map \(\Phi : \Omega \to \Omega\), a finite sequence \([z_1, \dots, z_k]\) of elements of \(\Omega\) is called a cycle of \(\Phi\) of length \(k\) if \(z_1, \dots, z_k\) are distinct, \(\Phi (z_i) = z_{i + 1}\) for \(i = 1, \dots, k - 1\), and \(\Phi (z_k) = z_1\). In this note it is shown that if \(\Omega\) is the free \(F\)-module of rank \(N\), where \(R\) is a finitely generated commutative domain of characteristic zero, and \(\Phi\) is a polynomial map, then the length \(k\) of cycles of \(\Phi\) is bounded by a number depending only on \(R\) and \(N\). This generalises earlier results for fields and rings of integers in algebraic number fields [see for example \textit{W. Narkiewicz}, Colloq. Math. 58, No. 1, 151-155 (1989; Zbl 0703.12002)].
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 757758
zbMATH Keywords: length of cycles of polynomial map
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
cites work: Polynomial mappings defined by forms with a common factor
cites work: Finiteness Properties of Polynomial Mappings
cites work: Invariant sets of morphisms on projective and affine number spaces
cites work: On polynomial transformations
cites work: On polynomial transformations in several variables
cites work: Polynomial cycles in algebraic number fields
cites work: Periodic points on an algebraic variety
cites work: Cycles of polynomial mappings in several variables
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description: scientific article; zbMATH DE number 757758