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.