On Artin's conjecture for pairs of diagonal forms
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866916921474023424 |
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| author | Campos-Vargas, João |
| author_facet | Campos-Vargas, João |
| contents | Let $p$ be an odd prime and $d = p^τ(p-1)$. In the spirit of Aritn's conjecture, consider the system of two diagonal forms of degree $d$ in $s$ variables given by \begin{equation*}\begin{split} a_1x_1^d + \cdots + a_sx_s^d = 0\\ b_1x_1^d + \cdots + b_sx_s^d = 0 \end{split} \end{equation*} with $a_i, b_i \in \mathbb{Q}_p$. For $s > 2 \frac{p}{p-1}d^2 - 2d$, this paper shows that this system has a non-trivial $p$-adic solution for every $τ\ge 3, p \ge 7$, and for every $τ= 2, p \ge \frac{C}{2}+4$, where $C \le 9996$. Moreover, for $s > (2\frac{p}{p-1} + \frac{C-3}{2p-2})d^2 - 2d$, this system will have a non-trivial $p$-adic solution for every $τ= 1, p \ge 5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_13024 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On Artin's conjecture for pairs of diagonal forms Campos-Vargas, João Number Theory Let $p$ be an odd prime and $d = p^τ(p-1)$. In the spirit of Aritn's conjecture, consider the system of two diagonal forms of degree $d$ in $s$ variables given by \begin{equation*}\begin{split} a_1x_1^d + \cdots + a_sx_s^d = 0\\ b_1x_1^d + \cdots + b_sx_s^d = 0 \end{split} \end{equation*} with $a_i, b_i \in \mathbb{Q}_p$. For $s > 2 \frac{p}{p-1}d^2 - 2d$, this paper shows that this system has a non-trivial $p$-adic solution for every $τ\ge 3, p \ge 7$, and for every $τ= 2, p \ge \frac{C}{2}+4$, where $C \le 9996$. Moreover, for $s > (2\frac{p}{p-1} + \frac{C-3}{2p-2})d^2 - 2d$, this system will have a non-trivial $p$-adic solution for every $τ= 1, p \ge 5$. |
| title | On Artin's conjecture for pairs of diagonal forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2009.13024 |