Finite polynomial orbits in finitely generated domains
type: Item
instance of: scholarly article
title: Finite polynomial orbits in finitely generated domains
zbMATH Open document ID: 0897.13024
DOI: 10.1007/BF01319041
published in: Monatshefte für Mathematik
publication date: 1998-06-10T00:00:00Z
full work available at URL: https://eudml.org/doc/178792
review text: Let \(R\) be a commutative domain of characteristic \(0\). A polynomial orbit in \(R\) is a set \(O_f(a)=\{ a_j: j=0,1,2,\ldots\}\), where \(a_0\in R\) and \(a_{j+1}=f(a_j)\) for \(j=0,1,2,\ldots\), for some non-zero polynomial \(f\in R[X]\). Such an orbit is called finite of length \(k\) if there are indices \(j<k\) with \(a_j=a_k\) and if \(k\) with this property is minimal. Such an orbit is called a polynomial cycle of length \(k\) if there is an index \(k>0\) with \(a_k=a_0\) and if \(k\) is minimal. Northcott showed that if \(R\) is the ring of integers of a number field, then for given \(f\), if an orbit \(O_f(a)\) has finite length, then this length is bounded above by a function of \(R\) and \(f\), and Narkiewicz showed that the dependence on \(f\) is unnecessary. The authors prove a more general and more precise result. Suppose that the domain \(R\) has the following two properties: (i) every polynomial cycle in \(R\) has length \(\leq B(R)\); (ii) the equation \(x_1+x_2+x_3=1\) has only finitely many solutions with \(x_i\in R^*\) and \(x_i\not= 1\) for \(i=1,2,3\), where \(R^*\) denotes the group of invertible elements of \(R\). Denote this number of solutions by \(C(R)\). It is known that such quantities \(B(R)\) and \(C(R)\) exist for every domain \(R\) that is finitely generated over \({\mathbb{Z}}\) and for several rings \(R\), explicit upper bounds for \(B(R)\) have been computed [cf. \textit{F. Halter-Koch} and \textit{W. Narkiewicz}, Monatsh. Math. 119, No. 4, 275-279 (1995; Zbl 0840.13003) for \(B(R)\) and \textit{H. P. Schlickewei}, Invent. Math. 102, No. 1, 95-107 (1990; Zbl 0711.11017) and \textit{J.-H. Evertse}, Invent. Math. 122, No. 3, 559-601 (1995; Zbl 0851.11019) for \(C(R)\)]. The authors show the following general result: \(D(R)\leq {1\over 3}(31+C(R))-1\). Using the explicit bounds mentioned above, the authors show among others that \(D({\mathbb{Z}})=4\) and that \(D(R_K)\leq {2\over 3}d4^d(31+2^{1031d})\) if \(R_K\) is the ring of inte…
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 11S05
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 1106742
zbMATH Keywords: polynomial orbits
zbMATH Keywords: finitely generated domains
zbMATH Keywords: polynomial cycle
author: Władysław Narkiewicz
author: Tadeusz Pezda
reviewed by: Jan-Hendrik Evertse
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1106742