# 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.

## Key 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.