On Multiplicative Independence of Rational Iterates

Marley Young

Published 2017-08-02Version 1

Lower bounds are given for the degree of multiplicative combinations of iterates of certain classes of rational functions over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao's method for constructing elements in $\mathbb{F}_{q^n}$ whose orders are larger than any polynomial in $n$ when $n$ becomes large. Additionally, for a field $\mathbb{F}$ of characteristic $0$, an upper bound is given for the number of polynomials $u \in \mathbb{F}[X]$ such that $\{ F_i(X,u(X)) \}_{i=1}^n$ is multiplicatively dependent for given rational functions $F_1,\ldots,F_n \in \mathbb{F}(X,Y)$.