Irreducible polynomials from a cubic transformation

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 $h(x{)}^{\mathrm{deg}f}\cdot f(R(x))$ 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.

20 pages