Polynomial cycles in rings of integers in fields of signature (\(0,2\)) (Q2637433)

Let \(R\) be a commutative ring with unity and \(f \in R[X]\), we define a cycle for \(f\) as a \(k\)-tuple \(x_{0}\), \(x_{1}\),\dots, \(x_{k-1}\) of different elements of \(R\) such that \[ f(x_{0}) =x_{1},\,\,\, f(x_{1}) =x_{2}, \,\,\, \dots,\,\,\, f(x_{k-1}) =x_{0}. \] The number \(k\) is called the length of the cycle. For an algebraic number field \(K\) we denote by \(Z_{K}\) the ring of algebraic integers in \(K\). In this paper, the author gives all possible cycle-lengths of polynomial mappings \(f \in Z_{K}[X]\) in the case where \(K\) is of signature \((0,2)\). In particular, he founds that the possible lengths are \(1, 2, 3, 4, 5, 6, 8, 10, 12.\) The cases of fields of signatures \((2,0)\), \( (0,1)\) or \((1,1)\) were treated earlier by other authors where they had found that the possible lengths are \(1, 2, 3, 4, 5, 6\).