On the irreducibility of $f(2^n,3^m,X)$ and other such polynomials
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| Format: | Preprint |
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2024
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| _version_ | 1866929336895930368 |
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| author | Bary-Soroker, Lior Garzoni, Daniele Matei, Vlad |
| author_facet | Bary-Soroker, Lior Garzoni, Daniele Matei, Vlad |
| contents | Let $f(t_1, \ldots, t_r, X)\in \mathbb{Z}[t_1, \ldots, t_r,X]$ be irreducible and let $a_1, \ldots, a_r\in \mathbb{Z} \smallsetminus \{0,\pm 1\}$. Under a necessary ramification assumption on $f$, and conditionally on the Generalized Riemann Hypothesis, we show that for almost all integers $n_1, \ldots, n_r$, the polynomial $f(a_1^{n_1}, \ldots, a_r^{n_r}, X)$ is irreducible in $\mathbb{Q}[X]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the irreducibility of $f(2^n,3^m,X)$ and other such polynomials Bary-Soroker, Lior Garzoni, Daniele Matei, Vlad Number Theory 11R09, 12E25, 14G05, 11A07 Let $f(t_1, \ldots, t_r, X)\in \mathbb{Z}[t_1, \ldots, t_r,X]$ be irreducible and let $a_1, \ldots, a_r\in \mathbb{Z} \smallsetminus \{0,\pm 1\}$. Under a necessary ramification assumption on $f$, and conditionally on the Generalized Riemann Hypothesis, we show that for almost all integers $n_1, \ldots, n_r$, the polynomial $f(a_1^{n_1}, \ldots, a_r^{n_r}, X)$ is irreducible in $\mathbb{Q}[X]$. |
| title | On the irreducibility of $f(2^n,3^m,X)$ and other such polynomials |
| topic | Number Theory 11R09, 12E25, 14G05, 11A07 |
| url | https://arxiv.org/abs/2405.04058 |