Anisotropic quadratic equations in three variables
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916695403134976 |
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| author | Li, Jiamin Liu, Jianya |
| author_facet | Li, Jiamin Liu, Jianya |
| contents | Let $f(x_1, x_2, x_3)$ be an indefinite anisotropic integral quadratic form with determinant $d(f)$, and $t$ a non-zero integer such that $d(f)t$ is square-free. It is proved in this paper that, as long as there is one integral solution to $f(x_1, x_2, x_3) = t$, there are infinitely many such solutions for which (i) $x_1$ has at most $6$ prime factors, and (ii) the product $x_1 x_2$ has at most $16$ prime factors. Various methods, such as algebraic theory of quadratic forms, harmonic analysis, Jacquet-Langlands theory, as well as combinatorics, interact here, and the above results come from applying the sharpest known bounds towards Selberg's eigenvalue conjecture. Assuming the latter the number $6$ or $16$ may be reduced to $5$ or $14$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anisotropic quadratic equations in three variables Li, Jiamin Liu, Jianya Number Theory 11P32, 11D09, 11N36 Let $f(x_1, x_2, x_3)$ be an indefinite anisotropic integral quadratic form with determinant $d(f)$, and $t$ a non-zero integer such that $d(f)t$ is square-free. It is proved in this paper that, as long as there is one integral solution to $f(x_1, x_2, x_3) = t$, there are infinitely many such solutions for which (i) $x_1$ has at most $6$ prime factors, and (ii) the product $x_1 x_2$ has at most $16$ prime factors. Various methods, such as algebraic theory of quadratic forms, harmonic analysis, Jacquet-Langlands theory, as well as combinatorics, interact here, and the above results come from applying the sharpest known bounds towards Selberg's eigenvalue conjecture. Assuming the latter the number $6$ or $16$ may be reduced to $5$ or $14$, respectively. |
| title | Anisotropic quadratic equations in three variables |
| topic | Number Theory 11P32, 11D09, 11N36 |
| url | https://arxiv.org/abs/2501.15033 |