Irreducible polynomials from a cubic transformation

TL;DR

This paper investigates the count of irreducible polynomials over finite fields formed via a cubic rational transformation, providing insights into their structure and applications in transformation shift registers.

Contribution

It introduces a method to count irreducible polynomials of a specific form involving cubic transformations, extending previous results on shift register irreducibility.

Findings

Derived a formula for counting irreducible polynomials with cubic transformations

Connected polynomial counts to irreducible transformation shift registers

Reproduced known results for shift register irreducibility

Abstract

Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field $\mathbb{F}_q$. We count the irreducible polynomials in $\mathbb{F}_q[x]$, of a given degree, which have the form $h(x)^{\mathrm{deg}\, f}\cdot f\bigl(R(x)\bigr)$ for some $f(x)\in\mathbb{F}_q[x]$. As an application, we recover the number of irreducible transformation shift registers of order three, previously computed by Jiang and Yang.