Prime divisors in polynomial orbits over function fields

Abstract

Given a polynomial ϕ over a global function field K / F q ( t ) and a wandering base point b ∈ K , we give a geometric condition on ϕ , ensuring the existence of primitive prime divisors for almost all points in the orbit O ϕ ( b ) : = { ϕ n ( b ) } n ⩾ 0 . As an application, we prove that the Galois groups (over K) of the iterates of many quadratic polynomials are large and use this to compute the density of prime divisors of O ϕ ( b ) .