Cycles of polynomial mappings in two variables over rings of integers in quadratic fields (Q1767464)

In a previous paper [Acta Arith. 108, 127--146 (2003; Zbl 1020.11066)] the author established a kind of Hasse principle for cycle-lengths for polynomial mappings \(R^N\longrightarrow R^N\), where \(N\geq2\), and \(R\) is the ring of integers of a finite extension of the rationals. Now he uses this principle to find a complete list of possible cycle-lengths of such mappings in the case when \(N=2\) and \(R\) is the ring of integers in a quadratic number field \(K=\mathbb Q(\sqrt d)\), where \(d\) is a square-free integer. There are \(14\) cases to be treated, the simplest occurring when \(d\equiv1\pmod 8\), in which case the set of cycle-lengths \(C(d)\) equals the smallest set closed under taking positive divisors and containing \(16, 18\) and \(24\). For comparison, if \(d\equiv797\pmod{840}\), then \(C(d)\) equals the smallest set closed under taking positive divisors and containing one of \(14\) integers, the largest being \(480\).