Full Galois groups of polynomials with slowly growing coefficients
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910765531791360 |
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| author | Bary-Soroker, Lior Goldgraber, Noam |
| author_facet | Bary-Soroker, Lior Goldgraber, Noam |
| contents | Choose a polynomial $f$ uniformly at random from the set of all monic polynomials of degree $n$ with integer coefficients in the box $[-L,L]^n$. The main result of the paper asserts that if $L=L(n)$ grows to infinity, then the Galois group of $f$ is the full symmetric group, asymptotically almost surely, as $n\to \infty$.
When $L$ grows rapidly to infinity, say $L>n^7$, this theorem follows from a result of Gallagher. When $L$ is bounded, the analog of the theorem is open, while the state-of-the-art is that the Galois group is large in the sense that it contains the alternating group (if $L< 17$, it is conditional on the general Riemann hypothesis). Hence the most interesting case of the theorem is when $L$ grows slowly to infinity.
Our method works for more general independent coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13559 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Full Galois groups of polynomials with slowly growing coefficients Bary-Soroker, Lior Goldgraber, Noam Number Theory 11R32, 12F10, 11R09, 11T23, 11T55 Choose a polynomial $f$ uniformly at random from the set of all monic polynomials of degree $n$ with integer coefficients in the box $[-L,L]^n$. The main result of the paper asserts that if $L=L(n)$ grows to infinity, then the Galois group of $f$ is the full symmetric group, asymptotically almost surely, as $n\to \infty$. When $L$ grows rapidly to infinity, say $L>n^7$, this theorem follows from a result of Gallagher. When $L$ is bounded, the analog of the theorem is open, while the state-of-the-art is that the Galois group is large in the sense that it contains the alternating group (if $L< 17$, it is conditional on the general Riemann hypothesis). Hence the most interesting case of the theorem is when $L$ grows slowly to infinity. Our method works for more general independent coefficients. |
| title | Full Galois groups of polynomials with slowly growing coefficients |
| topic | Number Theory 11R32, 12F10, 11R09, 11T23, 11T55 |
| url | https://arxiv.org/abs/2404.13559 |