On multiplicative independence of rational function iterates

TL;DR

This paper establishes lower bounds for the degree of multiplicative combinations of rational function iterates over fields, proving their multiplicative independence and extending Gao's method for constructing elements with large orders in finite fields.

Contribution

It provides new lower bounds for degrees of multiplicative combinations of rational iterates and generalizes Gao's method for finite field element construction.

Findings

Proves multiplicative independence of rational function iterates under certain conditions.

Establishes lower bounds for degrees of multiplicative combinations.

Discusses finiteness of polynomials translating finite sets to multiplicative dependence.

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.