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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2409.08625 |
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| _version_ | 1866912026495811584 |
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| author | Dubickas, Artūras Sha, Min |
| author_facet | Dubickas, Artūras Sha, Min |
| contents | In this paper, for positive integers $H$ and $k \leq n$, we obtain some estimates on the cardinality of the set of monic integer polynomials of degree $n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These include lower and upper bounds in terms of $H$ for fixed $k$ and $n$. We also count reducible and irreducible polynomials in that set separately. Our results imply, for instance, that the number of monic integer irreducible polynomials of degree $n$ and height at most $H$ whose all $n$ roots have equal moduli is approximately $2H$ for odd $n$, while for even $n$ there are more than $H^{n/8}$ of such polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08625 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting integer polynomials with several roots of maximal modulus Dubickas, Artūras Sha, Min Number Theory In this paper, for positive integers $H$ and $k \leq n$, we obtain some estimates on the cardinality of the set of monic integer polynomials of degree $n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These include lower and upper bounds in terms of $H$ for fixed $k$ and $n$. We also count reducible and irreducible polynomials in that set separately. Our results imply, for instance, that the number of monic integer irreducible polynomials of degree $n$ and height at most $H$ whose all $n$ roots have equal moduli is approximately $2H$ for odd $n$, while for even $n$ there are more than $H^{n/8}$ of such polynomials. |
| title | Counting integer polynomials with several roots of maximal modulus |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.08625 |