Cycle-lengths of a class of monic binomials
This paper is a contribution to research on the Morton-Silverman uniform boundedness conjecture in arithmetic dynamics. This conjecture states that the number of pre-periodic points of a rational map in an algebraic number field is bounded by a constant depending only on the dimension of the space, the degree of the map, and the degree of the field. Very little in known about this conjecture, even in one dimension; weaker conjectures have been formulated on bounds for the period of periodic orbits. The author considers the one-dimensional map \(f(x)=x^n+c\) over an algebraic number field \(K\) of degree \(N\), where the degree \(n\) of the map is the power of a prime \(p\). He proves that if \(K\) does not contain \(p\)-th primitive roots of unity, then the period of the periodic orbits of \(f\) in \(K\) is bounded by a constant depending only on \(K\) and \(p\), and if \(p>2^N\), this constant depends only on \(N\).
scientific article; zbMATH DE number 5758463
scientific article