A note on the stability of trinomials over finite fields

TL;DR

This paper investigates the stability of trinomials over finite fields, demonstrating that certain classes of these polynomials are not stable, especially in characteristic two and some odd characteristics, contributing to a conjecture on polynomial instability.

Contribution

It proves that even-degree trinomials over binary fields are not stable and extends similar results to specific monic trinomials over odd characteristic finite fields.

Findings

Even degree trinomials over 4 fields are not stable.

Certain monic trinomials over finite fields of odd characteristic are also unstable.

Supports a conjecture on the instability of polynomials with degrees divisible by the field's characteristic.

Abstract

A polynomial $f(x)$ over a field $K$ is called stable if all of its iterates are irreducible over $K$. In this paper we study the stability of trinomials over finite fields. Specially, we show that if $f(x)$ is a trinomial of even degree over the binary field $\mathbb{F}_2$, then $f(x)$ is not stable. We prove a similar result for some families of monic trinomials over finite fields of odd characteristic. These results are obtained towards the resolution of a conjecture on the instability of polynomials over finite fields whose degrees are divisible by the characteristic of the underlying field.