Solubility of a resultant equation and applications
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908404717453312 |
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| author | Browning, Tim Chan, Stephanie |
| author_facet | Browning, Tim Chan, Stephanie |
| contents | The large sieve is used to estimate the density of integral quadratic polynomials $Q$, such that there exists an odd degree integral polynomial which has resultant $\pm 1$ with $Q$. Given a monic integral polynomial $R$ of odd degree, this is used to show that for almost all integral quadratic polynomials $Q$, there exists a prime $p$ such that $Q$ and $R$ share a common root in the algebraic closure of the finite field with $p$ elements. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09264 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solubility of a resultant equation and applications Browning, Tim Chan, Stephanie Number Theory 11R29 (11G50, 11N36, 11R11, 11R45) The large sieve is used to estimate the density of integral quadratic polynomials $Q$, such that there exists an odd degree integral polynomial which has resultant $\pm 1$ with $Q$. Given a monic integral polynomial $R$ of odd degree, this is used to show that for almost all integral quadratic polynomials $Q$, there exists a prime $p$ such that $Q$ and $R$ share a common root in the algebraic closure of the finite field with $p$ elements. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given. |
| title | Solubility of a resultant equation and applications |
| topic | Number Theory 11R29 (11G50, 11N36, 11R11, 11R45) |
| url | https://arxiv.org/abs/2411.09264 |