Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866909704093958144 |
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| author | Anderson, Theresa C. Gafni, Ayla Oliver, Robert J. Lemke Lowry-Duda, David Shakan, George Zhang, Ruixiang |
| author_facet | Anderson, Theresa C. Gafni, Ayla Oliver, Robert J. Lemke Lowry-Duda, David Shakan, George Zhang, Ruixiang |
| contents | We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$ with $\mathrm{Gal}(f) \subseteq A_n$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $n$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $n$ number fields with almost prime discriminants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_02914 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants Anderson, Theresa C. Gafni, Ayla Oliver, Robert J. Lemke Lowry-Duda, David Shakan, George Zhang, Ruixiang Number Theory 11R45, 11N36, 11C08, 12E05 We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$ with $\mathrm{Gal}(f) \subseteq A_n$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $n$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $n$ number fields with almost prime discriminants. |
| title | Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants |
| topic | Number Theory 11R45, 11N36, 11C08, 12E05 |
| url | https://arxiv.org/abs/2107.02914 |