Number of Irreducible Polynomials and Pairs of Relatively Prime Polynomials in Several Variables over Finite Fields

TL;DR

This paper provides recursive formulas and asymptotic estimates for counting irreducible and relatively prime polynomial pairs in multiple variables over finite fields, extending previous results with improved formulas.

Contribution

It introduces recursive methods for counting irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields, generalizing prior work.

Findings

Number of irreducibles computed recursively by degree

Number of relatively prime pairs expressed via irreducibles

Asymptotic formulas generalize and improve previous results

Abstract

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the {\em vector degree}, are considered. We show that the number of irreducibles can be computed recursively by degree and that the number of relatively prime pairs can be expressed in terms of the number of irreducibles. We also obtain asymptotic formulas for the number of irreducibles and the number of relatively prime pairs. The asymptotic formulas for the number of irreducibles generalize and improve several previous results by Carlitz, Cohen and Bodin.