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arXiv · 2601.12968

Counting Irreducible polynomials with coefficients from thin subgroups

Alina Ostafe·Igor E. Shparlinski
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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).

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Alina Ostafe, Igor E. Shparlinski. 2026-08-01. Counting Irreducible polynomials with coefficients from thin subgroups. https://arxiv.org/abs/2601.12968

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