Stability of Certain Higher Degree Polynomials

TL;DR

This paper investigates the stability of specific higher degree polynomials over fields, demonstrating conditions under which their iterates remain irreducible, with particular focus on polynomials of the form z^d + 1/c.

Contribution

It establishes new results on the irreducibility and stability of the polynomial family f(z)=z^d+1/c for various degrees and parameters, extending understanding in arithmetic dynamics.

Findings

For infinite families with d≥3, irreducibility implies stability.

All iterates of z^2+1/c are irreducible when c ≡ 1 mod 4.

For d=3, reducible polynomials have exactly 2 irreducible factors for |c| ≤ 10^12.

Abstract

One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. In this paper, we study the stability of $f(z)=z^d+\frac{1}{c}$ for $d\geq 2$, $c\in{\mathbb{Z}\setminus\{0\}}$. We show that for infinite families of $d\geq 3$, whenever $f(z)$ is irreducible, all its iterates are irreducible, that is, $f(z)$ is stable. For $c\equiv 1\pmod{4}$, we show that all the iterates of $z^2+\frac{1}{c}$ are irreducible. Also we show that for $d=3$, if $f(z)$ is reducible, then the number of irreducible factors of each iterate of $f(z)$ is exactly $2$ for $|c|\leq{10^{12}}$.