Irreducible Polynomials with Varying Constraints on Coefficients

TL;DR

This paper investigates the distribution of prime polynomials over finite fields with various coefficient constraints, extending previous work on digit restrictions and providing asymptotic counts for such primes.

Contribution

It introduces a framework for counting prime polynomials with diverse coefficient restrictions, generalizing prior results and establishing new asymptotic formulas.

Findings

Derived asymptotic formulas for prime polynomials with coefficient constraints

Extended previous digit restriction results to polynomial coefficients

Connected function field analogues with integer prime digit problems

Abstract

We study the number of prime polynomials of degree $n$ over $\mathbb{F}_q$ in which the $i^{th}$ coefficient is either preassigned to be $a_i \in \mathbb{F}_q$ or outside a small set $S_i \subset \mathbb{F}_q$. This serves as a function field analogue of a recent work of Maynard, which counts integer primes that do not have specific digits in their base-$q$ expansion. Our work relates to Pollack's and Ha's work, which count the amount of prime polynomials with $\ll \sqrt{n}$ and $\ll n$ preassigned coefficients, respectively. Our result demonstrates how one can prove asymptotics of the number of prime polynomials with different types of constraints to each coefficient.