×

An algorithm determining the set of lengths of polynomial cycles in \(Z_K^N\). (English) Zbl 1480.11136

For an integer \(N\) let \(\Phi_1,\dots,\Phi_N\) be polynomials over a commutative ring \(R\) and denote by \(CYCL(R,N)\) the set of cardinalities of finite cyclic orbits of the mapping \(\Phi:\ R^N\longrightarrow R^N\) defined by \[ \Phi(\bar x_1,\dots,\bar x_N)=(\Phi_1(\bar x_1,\dots,\bar x_N),\dots, \Phi_N(\bar x_1,\dots,\bar x_N)). \] In Theorem 1.1 the author presents an algorithm to determine \(CYCL(R,N)\) in the case when \(R\) is a discrete valuation ring of characteristic zero. A complete description of \(CYCL(R,N)\) when \(R\) is a discrete valuation ring which is either unramified or of positive characteristic has been given by the author earlier [Commun. Math. 21, No. 2, 129–135 (2013; Zbl 1294.37032); Mich. Math. J. 64, No. 1, 109–142 (2015; Zbl 1317.13047)]. In the second paper also a description of \(CYCL(Z,N)\) has been given.
The application of the Hasse principle for \(CYCL(R,N)\) established by the author earlier (Theorem 3.2 in [Acta Arith. 108, No. 2, 127–146 (2003; Zbl 1020.11066)]) leads to an algorithm for \(CYCL(Z_K,N)\), where \(N\ge2\) and \(Z_K\) is the ring of integers of an algebraic number field \(K\) of finite degree (Theorem 1.2).
An algorithm for \(CYCL(Z_K,1)\) has been given by the author in an earlier paper [Publ. Math. 84, No. 3–4, 399–414 (2014; Zbl 1324.11061)] with an essentially different proof. Complete lists of elements of \(CYCL(Z_K,1)\) are known for quadratic fields [J. Boduch, MA thesis. Wrocław University (1990)] cubic fields of negative discriminant [the reviewer, Funct. Approximatio, Comment. Math. 35, 261–269 (2006; Zbl 1196.37130)] and totally imaginary quartic fields [the author, Funct. Approximatio, Comment. Math. 49, No. 2, 391–409 (2013; Zbl 1285.11128)]. A list of elements of \(CYCL(Z_K,2)\) for quadratic fields \(K\) has been given by the author in [Cent. Eur. J. Math. 2, No. 2, 294–331 (2004; Zbl 1126.11057)].

MSC:

11R09 Polynomials (irreducibility, etc.)
11C08 Polynomials in number theory
11S05 Polynomials
11S82 Non-Archimedean dynamical systems
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F30 Valuation rings
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P15 Dynamical systems over global ground fields
37P35 Arithmetic properties of periodic points
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Boduch, MA thesis, Univ. of Wrocław, 1990.
[2] H. Cohen,A Course in Computational Algebraic Number Theory, Grad. Texts in Math. 138, Springer, 1993. · Zbl 0786.11071
[3] H. Cohen,Advanced Topics in Computational Number Theory, Grad. Texts in Math. 193, Springer, 2000. · Zbl 0977.11056
[4] B. Hutz,Dynatomic cycles for morphisms of projective varieties, New York J. Math. 16 (2010), 125-159. · Zbl 1229.14022
[5] W. Narkiewicz,Polynomial cycles in certain rings of rationals, J. Théorie Nombres Bordeaux 14 (2002), 529-552. · Zbl 1071.11017
[6] W. Narkiewicz,Polynomial cycles in cubic fields of negative discriminant, Funct. Approx. Comment. Math. 35 (2006), 261-269. · Zbl 1196.37130
[7] T. Pezda,Cycles of polynomial mappings in several variables over rings of integers in finite extensions of the rationals, Acta Arith. 108 (2003), 127-146. · Zbl 1020.11066
[8] T. Pezda,Cycles of polynomial mappings in two variables over rings of integers in quadratic fields, Centr. Eur. J. Math. 2 (2004), 294-331. · Zbl 1126.11057
[9] T. Pezda,Cycles of polynomial mappings in several variables over rings of integers in finite extensions of the rationals, II, Monatsh. Math. 145 (2005), 321-331. · Zbl 1197.37143
[10] T. Pezda,On some issues concerning polynomial cycles, Comm. Math. 21 (2013), 129-135. · Zbl 1294.37032
[11] T. Pezda,Polynomial cycles in rings of integers in fields of signature(0,2), Funct. Approx. Comment. Math. 49 (2013), 391-409. · Zbl 1285.11128
[12] T. Pezda,An algorithm determining cycles of polynomial mappings in integral domains, Publ. Math. Debrecen 84 (2014), 399-414. · Zbl 1324.11061
[13] T. Pezda,Cycles of polynomial mappings in several variables over discrete valuation rings and over Z, Michigan Math. J. 64 (2015), 109-142. · Zbl 1317.13047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.