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 Fq. We count the irreducible polynomials in Fq[x], of a given degree, that have the form h(x)deg⁡f⋅f(R(x)) for some f(x)∈Fq[x]. As an example of application of our results, we recover the number of irreducible transformation shift registers of order three, which were computed by Jiang and Yang in 2017.