论文标题
关于Artin对对角线形式的猜想
On Artin's conjecture for pairs of diagonal forms
论文作者
论文摘要
令$ p $为奇数，$ d = p^τ（p-1）$。本着Aritn的猜想的精神，请考虑\ S $变量中两种对角度的$ d $的系统系统，由\ begin {equation*} \ begin {split} a_1x_1^d + \ cdots + a_sx_s + a_sx_s + a_sx_s^d = 0 \ end {equation*}带有$ a_i，b_i \ in \ mathbb {q} _p $。对于$ s> 2 \ frac {p} {p-1} d^2-2d $，本文表明，对于每个$τ\ ge 3，p \ ge 7 $，对于每个$τ\ ge 7 $，并且每个$τ= 2，p \ ge 7 $，以及每个$τ= 2，p \ ge \ ge \ ge \ ge \ frac {c} {c} {2}+4 $ c \ pe c。此外，对于$ s>（2 \ frac {p} {p-1} + \ frac {c-3} {2p-2}）d^2-2d $，此系统将具有每个$ p $ -Adic-adic解决方案的每个$τ= 1，p \ ge 5 $。
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 $τ= 1, p \ge 5$.
