An algorithm determining cycles of polynomial mappings in integral domains (Q2875442)

If \(R\) is an integral domain and \(f\in R[X]\), then a sequence \(S: x_0,\dots,x_{n-1}\) of distinct elements of \(R\) is called a cycle of \(f\) if for \(i=0,1,\dots,n-2\) one has \(f(x_i)=x_{i+1}\) and \(f(x_{n-1})=x_0\). A cycle \(S\) is called normalized if \(x_0=0,x_1=1\), and to find all polynomial cycles in \(R\) it suffices to determine all normalized cycles. The author proves that if all unit solutions of the equations NEWLINE\[NEWLINEu+v=1\;\text{and}\;u+v+w=1\quad (u,v,w\neq1)\tag{1}NEWLINE\]NEWLINE are known, then there is an algorithm leading to a list of all normalized polynomial cycles in \(R\). This covers in particular all finitely generated domains of zero characteristic.NEWLINENEWLINE It has been known earlier under stronger assumptions (for every non-zero \(b\in R\) the equations \(u+bv=1\), \(b(u+v)+w=1\) (with \(w\neq1\)) have only finitely many unit solutions, and the equation \(u_1+u_2+\cdots+u_5=1\) has finitely many unit solutions without vanishing subsums of the left hand-side) that there only finitely many normalized polynomial cycles (\textit{F. Halter-Koch} and the reviewer [Publ. Math., 56, No. 3-4, 405--414; (2000; Zbl 0961.11005)]).NEWLINENEWLINE In the case when \(R=Z_K\) is the ring of integers of a finite extension \(K\) of the rationals there is an efficient algorithm for solutions of the first equation in (1) (\textit{K. Wildanger} [J. Number Theory, 82, No. 2, 188--224 (2000; Zbl 0952.11032)], but no such algorithm is known for the second equation (except the easy case when the unit rank of the field equals \(1\)). Nevertheless the author gives in Theorem 2 a procedure to find all normalized polynomial cycles in \(Z_K\) using only the solutions of the first equation in (1). In Theorem 3 the same is done for finite nonlinear orbits of polynomials in \(Z_K\) (an orbit is nonlinear if it cannot be realized by a linear polynomial). It is shown in particular that the cardinalities of these orbits are bounded by a constant depending only of the degree of the field \(K\).