Polynomial cycles in finitely generated domains
For a non-empty set \(\Omega\) and a map \(\Phi : \Omega \to \Omega\), a finite sequence \([z_1, \dots, z_k]\) of elements of \(\Omega\) is called a cycle of \(\Phi\) of length \(k\) if \(z_1, \dots, z_k\) are distinct, \(\Phi (z_i) = z_{i + 1}\) for \(i = 1, \dots, k - 1\), and \(\Phi (z_k) = z_1\). In this note it is shown that if \(\Omega\) is the free \(F\)-module of rank \(N\), where \(R\) is a finitely generated commutative domain of characteristic zero, and \(\Phi\) is a polynomial map, then the length \(k\) of cycles of \(\Phi\) is bounded by a number depending only on \(R\) and \(N\). This generalises earlier results for fields and rings of integers in algebraic number fields [see for example \textit{W. Narkiewicz}, Colloq. Math. 58, No. 1, 151-155 (1989; Zbl 0703.12002)].
scientific article; zbMATH DE number 757758