On multiplicative independence of rational function iterates

Marley Young

doi:10.1007/s00605-020-01393-9

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.

Highlights

Introduction and main resultsWe say that n non-zero elements a1, . . . , an of a ring are multiplicatively independent if, for integers k1, . . . , kn, we have that a1k1 . . . ankn = 1 if and only if k1 = · · · = kn = 0
As such, [4], which leads into the area of “unlikely intersections”, really concerns the multiplicative dependence of points on curves
To study the multiplicative independence of elements in the orbits of polynomials or rational functions, it is necessary to know when the given functions are multiplicatively dependent, as in this case all their values must be multiplicatively dependent. We study this problem in the context of iterates of rational functions over a field

Summary

Introduction

Theorem 1.5 Suppose F is a field of characteristic zero, n is a positive integer, and Fi = Gi /Hi ∈ F(X , Y ) are rational functions for 1 ≤ i ≤ n, of respective degrees d1 ≤ · · · ≤ dn in X and 1 ≤ e1 ≤ · · · ≤ en in Y . Lemma 2.4 Throughout, if min{μ, ν} < ∞, define δ = | deg gmin{μ,ν} − deg hmin{μ,ν}|, and for a positive integer j, let S j and Tj be respectively the degrees of the lowest order term in g j and h j .

Objectives

Results

Conclusion