Title: On multiplicative independence of rational function iterates,
Abstract: We give lower bounds for the degree of multiplicative combinations of
iterates of rational functions (with certain exceptions) over a general field,
establishing the multiplicative independence of said iterates. This leads to a
generalisation of Gao's method for constructing elements in the finite field
$\mathbb{F}_{q^n}$ whose orders are larger than any polynomial in $n$ when $n$
becomes large. Additionally, we discuss the finiteness of polynomials which
translate a given finite set of polynomials to become multiplicatively
dependent.