Iteration of Polynomials $AX^d+C$ Over Finite Fields

TL;DR

This paper investigates the behavior of polynomial iterations over finite fields, establishing asymptotic formulas for the size of the image set after multiple iterations under specific divisibility conditions.

Contribution

It provides a new asymptotic estimate for the size of iterated polynomial images over finite fields when the degree divides p-1.

Findings

Asymptotic formula for the size of iterated polynomial images

Conditions under which the formula holds (d | p-1)

Behavior of polynomial iterations with distinct orbit points

Abstract

For a polynomial $f(X)=AX^d+C \in \mathbb{F}_p[X]$ with $A\neq 0$ and $d\geq 2$, we prove that if $d\;|\;p-1$ and $f^i(0)\neq f^j(0)$ for $0\leq i<j\leq N$, then $\#f^N(\mathbb{F}_p) \sim \frac{2p}{(d-1)N},$ where $f^N$ is the $N$-th iteration of $f$.