Irreducible polynomials over a finite field with restricted coefficients

TL;DR

This paper establishes an asymptotic count for irreducible polynomials over finite fields with coefficients restricted to a subset, extending Maynard's prime digit restriction results to function fields.

Contribution

It provides a novel asymptotic formula for irreducible polynomials with restricted coefficients over finite fields, analogous to prime digit restrictions in number theory.

Findings

Asymptotic formula for the count of such polynomials

Extension of Maynard's prime digit restriction results to function fields

Conditions on parameters n and q for the asymptotic to hold

Abstract

We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a finite field F_q whose coefficients are restriced to lie in a given subset of F_q.