Least zero of pairs of additive cubic equations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917285016371200 |
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| author | Xiao, Yixiu Li, Hongze |
| author_facet | Xiao, Yixiu Li, Hongze |
| contents | An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms \[ \begin{cases} F = c_{1}x_1^3 + c_{2}x_2^3 + \cdots + c_{n}x_n^3 = 0, \\ G = d_{1}x_1^3 + d_{2}x_2^3 + \cdots + d_{n}x_n^3 = 0, \end{cases} \] under the "$M$-good" condition for $n \ge 16$, where $c_{1}, \dots, c_{n}$ and $d_{1}, \dots, d_{n}$ are integers. Additionally, a range is derived for the probability that randomly selected simultaneous equations satisfy the $M$-good condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10001 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Least zero of pairs of additive cubic equations Xiao, Yixiu Li, Hongze Number Theory 11D72, 11D25, 11P55 An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms \[ \begin{cases} F = c_{1}x_1^3 + c_{2}x_2^3 + \cdots + c_{n}x_n^3 = 0, \\ G = d_{1}x_1^3 + d_{2}x_2^3 + \cdots + d_{n}x_n^3 = 0, \end{cases} \] under the "$M$-good" condition for $n \ge 16$, where $c_{1}, \dots, c_{n}$ and $d_{1}, \dots, d_{n}$ are integers. Additionally, a range is derived for the probability that randomly selected simultaneous equations satisfy the $M$-good condition. |
| title | Least zero of pairs of additive cubic equations |
| topic | Number Theory 11D72, 11D25, 11P55 |
| url | https://arxiv.org/abs/2510.10001 |