Generating series for irreducible polynomials over finite fields

TL;DR

This paper develops generating series and formulas to count irreducible polynomials over finite fields in multiple variables, providing exact and asymptotic results for various cases including multi-degree and indecomposability.

Contribution

It introduces new generating series and formulas for counting irreducible polynomials over finite fields in multiple variables, extending previous work.

Findings

Derived explicit formulas for counting irreducible polynomials

Provided asymptotic approximations for large degrees

Analyzed cases of multi-degree and indecomposable polynomials

Abstract

We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the case of the multi-degree and the case of indecomposable polynomials.