Binary forms with covariant points close to the real axis
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913512430764032 |
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| author | Rosu, Eugenia |
| author_facet | Rosu, Eugenia |
| contents | For a real binary form $F(X, Z)$, Stoll and Cremona have defined a reduction theory using the action of the modular group $SL_2(\mathbb{Z})$, and associated to each binary form a covariant point $z(F)$ located in the upper half plane. When the point $z(F)$ is close to the real axis, then at least half of the roots will be on a circle of small radius $r$. Conversely, we find conditions depending on the radius $r$ such that the covariant point $z(F)$ to be close to the real axis. The results have further applications to improving the reduction algorithm for binary forms of Stoll and Cremona. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14112 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Binary forms with covariant points close to the real axis Rosu, Eugenia Number Theory For a real binary form $F(X, Z)$, Stoll and Cremona have defined a reduction theory using the action of the modular group $SL_2(\mathbb{Z})$, and associated to each binary form a covariant point $z(F)$ located in the upper half plane. When the point $z(F)$ is close to the real axis, then at least half of the roots will be on a circle of small radius $r$. Conversely, we find conditions depending on the radius $r$ such that the covariant point $z(F)$ to be close to the real axis. The results have further applications to improving the reduction algorithm for binary forms of Stoll and Cremona. |
| title | Binary forms with covariant points close to the real axis |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.14112 |