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Counting Irreducible polynomials with coefficients from thin subgroups

Alina Ostafe, Igor E. Shparlinski

TL;DR

This work studies irreducible polynomials over finite fields $\mathbb{F}_q$ whose coefficients lie in thin subgroups, extending prior results that restricted coefficients to squares or finite unions of polynomial images. The authors develop a new algebraic-geometry–based approach using discriminant varieties and Weil-type bounds, enabling asymptotics for $\mathcal{I}_n(\mathbf{h},\mathbf{G})$ even when the index of the coefficient-subgroup grows as a power of $q$, and a uniform estimate for counts with prescribed discriminant $d$, $\mathcal{I}_{d,n}(\mathbf{h},\mathbf{G})$, including a discriminant-constrained generalization that Chebotarev-based methods struggle to achieve. They prove absolute irreducibility of discriminant hypersurfaces $\mathcal{D}_{d,n}$ in odd characteristic and bound intersections with derivative-residue varieties to control degeneracies, yielding power-saving error terms. The results apply to subgroups of index up to $q^{1/2-\varepsilon}$ and reveal a robust framework connecting finite-field irreducibility with discriminant geometry, offering new tools for coefficient-restriction problems in polynomial counting.

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).

Counting Irreducible polynomials with coefficients from thin subgroups

TL;DR

This work studies irreducible polynomials over finite fields whose coefficients lie in thin subgroups, extending prior results that restricted coefficients to squares or finite unions of polynomial images. The authors develop a new algebraic-geometry–based approach using discriminant varieties and Weil-type bounds, enabling asymptotics for even when the index of the coefficient-subgroup grows as a power of , and a uniform estimate for counts with prescribed discriminant , , including a discriminant-constrained generalization that Chebotarev-based methods struggle to achieve. They prove absolute irreducibility of discriminant hypersurfaces in odd characteristic and bound intersections with derivative-residue varieties to control degeneracies, yielding power-saving error terms. The results apply to subgroups of index up to and reveal a robust framework connecting finite-field irreducibility with discriminant geometry, offering new tools for coefficient-restriction problems in polynomial counting.

Abstract

L. Bary-Soroker and R. Shmueli (2026) have given an asymptotic formula for the number of irreducible polynomials over the finite fields of elements, such that their coefficients are perfect squares in 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 . 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).
Paper Structure (14 sections, 7 theorems, 38 equations)

This paper contains 14 sections, 7 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Our results
  4. Algebraic geometry tools
  5. Rational points on varieties
  6. Degrees of intersections
  7. Discriminant variety
  8. Irreducibility of the discriminant variety
  9. Counting polynomials with non-relatively prime derivatives and given discriminant
  10. Proof of Theorem \ref{['thm:IdhG']}
  11. Splitting polynomials into equivalence classes
  12. Estimating the contribution from good and bad equivalence classes
  13. Estimating the number of bad equivalence classes and concluding the proof
  14. Comments

Key Result

Theorem 1.1

Let $\mathbb{F}_q$ be of characteristic $p > n\geqslant 2$. For $d \in \mathbb{F}_q^*$, $\mathbf{h} \in \mathbb{F}_q^*$ and $\mathbf{G} \leqslant$F_q^*$^n$ we have

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 1 more