Irreducibility of random polynomials of $\mathbb{Z}[X]$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915390116855808 |
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| author | Bazin, Pierre-Alexandre |
| author_facet | Bazin, Pierre-Alexandre |
| contents | In a recent paper, Bary-Soroker, Koukoulopoulos and Kozma proved that when $A$ is a random monic polynomial of $\mathbb{Z}[X]$ of deterministic degree $n$ with coefficients $a_j$ drawn independently according to measures $μ_j,$ then $A$ is irreducible with probability tending to $1$ as $n\to\infty$ under a condition of near-uniformity of the $μ_j$ modulo four primes (which notably happens when the $μ_j$ are uniform over a segment of $\mathbb Z$ of length $N\ge 35.$) We improve here the speed of convergence when we have the near-uniformity condition modulo more primes. Notably, we obtain the optimal bound $\mathbb{P}(A \text{ irreducible}) = 1 -O(1/\sqrt n)$ when we have near-uniformity modulo twelve primes, which notably happens when the $μ_j$ are uniform over a segment of length $N\ge 10^8.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06643 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Irreducibility of random polynomials of $\mathbb{Z}[X]$ Bazin, Pierre-Alexandre Number Theory In a recent paper, Bary-Soroker, Koukoulopoulos and Kozma proved that when $A$ is a random monic polynomial of $\mathbb{Z}[X]$ of deterministic degree $n$ with coefficients $a_j$ drawn independently according to measures $μ_j,$ then $A$ is irreducible with probability tending to $1$ as $n\to\infty$ under a condition of near-uniformity of the $μ_j$ modulo four primes (which notably happens when the $μ_j$ are uniform over a segment of $\mathbb Z$ of length $N\ge 35.$) We improve here the speed of convergence when we have the near-uniformity condition modulo more primes. Notably, we obtain the optimal bound $\mathbb{P}(A \text{ irreducible}) = 1 -O(1/\sqrt n)$ when we have near-uniformity modulo twelve primes, which notably happens when the $μ_j$ are uniform over a segment of length $N\ge 10^8.$ |
| title | Irreducibility of random polynomials of $\mathbb{Z}[X]$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.06643 |