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On multiplicative dependence of values of rational functions and a generalization of the Northcott theorem. (English) Zbl 1442.11142
In this paper the authors study multiplicative dependence of values of polynomials or rational functions over a number field. Let \(K\) be a number field and let \(\phi= (\phi_1,\dots,\phi_s) \in K(X)^s\) whose components cannot multiplicatively generate a power of a linear fractional function. It is shown that then there are only finitely many elements \(\alpha\) in the maximal abelian extension of \(K\) such that \(\phi(\alpha)\) is a multiplicatively dependent point. The authors also obtain a generalization of the Northcott theorem replacing the finiteness of preperiodic points from a given number field by the finiteness of algebraic integers having two multiplicatively dependent elements in their orbits. Let \(\phi(X)\) be of degree \(d \geq 2\) and not of the form \(\beta X^{\pm d}\) with \(\beta \in K^*\), and let \(\alpha \in \overline{\mathbb Q} \setminus U\) (where \(U\) is the set of all roots of unity) be nonpreperiodic for \(\phi\). Then, there are only finitely many positive integers \(n\) such that \(\alpha\) and \(\phi^{(n)}(\alpha)\) (where \(\phi^{(n)}\) is the \(n\)th iteration of \(\phi\)) are multiplicatively dependent.

11R18 Cyclotomic extensions
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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