Counting Irreducible polynomials with coefficients from thin subgroups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914263510024192 |
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| author | Ostafe, Alina Shparlinski, Igor E. |
| author_facet | Ostafe, Alina Shparlinski, Igor E. |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12968 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting Irreducible polynomials with coefficients from thin subgroups Ostafe, Alina Shparlinski, Igor E. Number Theory 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). |
| title | Counting Irreducible polynomials with coefficients from thin subgroups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.12968 |