Riccati equations and polynomial dynamics over function fields

TL;DR

This paper investigates primitive prime divisors and Galois groups of polynomial iterates over function fields, providing new results especially in characteristic p and constructing explicit examples with full Galois groups.

Contribution

It proves finiteness of primitive prime divisors in backward orbits under weak conditions and constructs the first non-isotrivial polynomials with Galois groups of finite index.

Findings

Primitive prime divisors occur in all but finitely many orbit terms.

Constructed explicit non-isotrivial polynomials with maximal Galois groups.

Almost all quadratic polynomials over $\

Abstract

Given a function field $K$ and $\phi \in K[x]$, we study two finiteness questions related to iteration of $\phi$: whether all but finitely many terms of an orbit of $\phi$ must possess a primitive prime divisor, and whether the Galois groups of iterates of $\phi$ must have finite index in their natural overgroup $\text{Aut}(T_d)$, where $T_d$ is the infinite tree of iterated preimages of $0$ under $\phi$. We focus particularly on the case where $K$ has characteristic $p$, where far less is known. We resolve the first question in the affirmative under relatively weak hypotheses; interestingly, the main step in our proof is to rule out "Riccati differential equations" in backwards orbits. We then apply our result on primitive prime divisors and adapt a method of Looper to produce a family of polynomials for which the second question has an affirmative answer; these are the first non-isotrivial examples of such polynomials. We also prove that almost all quadratic polynomials over $\mathbb{Q}(t)$ have iterates whose Galois group is all of $\text{Aut}(T_d)$.