Prime divisors in polynomial orbits over function fields

TL;DR

This paper establishes geometric conditions under which primitive prime divisors appear in the orbits of polynomial dynamical systems over global function fields, extending understanding of prime divisors in such contexts.

Contribution

It introduces a geometric criterion that guarantees primitive prime divisors for almost all points in polynomial orbits over function fields, a novel result in arithmetic dynamics.

Findings

Primitive prime divisors exist for almost all orbit points under the given conditions.

The geometric condition relates to the polynomial's properties over the function field.

The results extend classical number theory concepts to the setting of function fields.

Abstract

Given a polynomial $\phi$ over a global function field $K$ and a wandering base point $b\in K$, we give a geometric condition on $\phi$ ensuring the existence of primitive prime divisors for almost all points in the orbit $\mathcal{O}_\phi(b):=\{\phi^n(b)\}_{n\geq0}$