##
**Finite polynomial orbits in finitely generated domains.**
*(English)*
Zbl 0897.13024

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. F. Halter-Koch and W. Narkiewicz, Monatsh. Math. 119, No. 4, 275-279 (1995; Zbl 0840.13003) for \(B(R)\) and H. P. Schlickewei, Invent. Math. 102, No. 1, 95-107 (1990; Zbl 0711.11017) and 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 integers of a number field \(K\) of degree \(d\). They give a more general result for rings of \(S\)-integers.

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. F. Halter-Koch and W. Narkiewicz, Monatsh. Math. 119, No. 4, 275-279 (1995; Zbl 0840.13003) for \(B(R)\) and H. P. Schlickewei, Invent. Math. 102, No. 1, 95-107 (1990; Zbl 0711.11017) and 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 integers of a number field \(K\) of degree \(d\). They give a more general result for rings of \(S\)-integers.

Reviewer: J.-H.Evertse (Leiden)

### MSC:

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

13B25 | Polynomials over commutative rings |

11S05 | Polynomials |

11R09 | Polynomials (irreducibility, etc.) |

13E15 | Commutative rings and modules of finite generation or presentation; number of generators |

13G05 | Integral domains |

PDF
BibTeX
XML
Cite

\textit{W. Narkiewicz} and \textit{T. Pezda}, Monatsh. Math. 124, No. 4, 309--316 (1997; Zbl 0897.13024)

### References:

[1] | Cohen H (1995) A Course in Computational Number Theory, 2nd edn. Berlin Heidelberg New York: Springer |

[2] | Cusick TW (1983) Lower bounds for regulators. In: Number Theory, Noordwijkerhout. Lect Notes Math 1068, pp. 63-73. Berlin Heidelberg New York: Springer |

[3] | Evertse JH (1995) The number of solutions of decomposable form equations. Invent Math122: 559-601 · Zbl 0851.11019 |

[4] | Evertse JH, Gy?ry K (1988) On the number of solutions of weighted unit equations. Compos Math66: 329-354 · Zbl 0644.10015 |

[5] | Halter-Koch F, Narkiewicz W (1995) Polynomial cycles in finitely generated domains. Mh Math119: 275-279 · Zbl 0840.13003 |

[6] | Narkiewicz W (1989) Polynomial cycles in algebraic number fields. Colloq Math58: 149-153 · Zbl 0703.12002 |

[7] | Nagell T (1959) Les points exceptionnels rationnels sur certaines cubiques du premier genre. Acta Arith5: 333-357 · Zbl 0093.04802 |

[8] | Nagell T (1969) Quelques probl?mes relatifs aux unit?s alg?briques. Arkiv f Mat8: 115-127 · Zbl 0213.06902 |

[9] | Northcott DG (1950) Periodic points on an algebraic variety. Ann Math51: 510-518 · Zbl 0036.30102 |

[10] | Pezda T (1994) Polynomial cycles in certain local domains. Acta Arith66: 11-22 · Zbl 0803.11063 |

[11] | Pezda T (1994) Cycles of polynomial mappings in several variables. Manuscripta Math83: 279-289 · Zbl 0804.11059 |

[12] | van der Poorten AJ, Schlickewei HP (1991) Additive relations in fields. J Austral Math Soc ser A51: 154-170 · Zbl 0747.11017 |

[13] | Samuel P (1966) ? propos du th?or?me des unit?s. Bull Sci Math (2)90: 84-96 · Zbl 0166.30701 |

[14] | Schlickewei HP (1990) S-units equations over number fields. Inv Math102: 95-107 · Zbl 0711.11017 |

[15] | Schilickewei HP Linear equations over finitely generated groups. Ann Math (to appear) |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.