Counting Irreducible polynomials with coefficients from thin subgroups

TL;DR

This paper introduces a new method to count irreducible polynomials over finite fields with coefficients from small subgroups, extending previous results and enabling additional conditions like prescribed discriminant values.

Contribution

The paper presents a novel approach for counting irreducible polynomials with coefficients in small subgroups, surpassing previous methods and allowing for new constraints such as fixed discriminant.

Findings

Successfully generalized counting to coefficients from small subgroups

Enabled counting with prescribed discriminant values

Provided a more versatile approach than prior work

Abstract

L. Bary-Soroker and R. Shmueli (2026) have given an asymptotic formula for the number of irreducible polynomials over the finite fields $\mathbb F_q$ of $q$ elements, such that their coefficients are perfect squares in $\mathbb F_q$ and also extended this to classes of polynomials with coefficients described by finitely many unions of intersections of polynomial images. Here we use a different approach, which allows us to obtain another generalisation of this result to polynomials with coefficients from small subgroups of $\mathbb F_q^*$. As a demonstration of the power of our approach, we also use it to count such irreducible polynomials with an additional condition, namely, with a prescribed value of their discriminant. This generalisation seems to be unachievable via the approach of L. Bary-Soroker and R. Shmueli (2026).