论文标题
关于$ \ mathbb {f} _q（t）$的Davenport和对角线立方体的问题
On a question of Davenport and diagonal cubic forms over $\mathbb{F}_q(t)$
论文作者
论文摘要
给定一个非明显的对角线超出表面$ x \ subset \ subset \ mathbb {p}^{n-1} $ a $ \ mathbb {f} _q（t）$，带有$ \ mathrm {charrm {char} $ $ o（| p |^{3+ \ varepsilon}）$ for $ n = 6 $和$ o（\ lvert p \ rvert^{2+ \ varepsilon}）$ for $ n = 4 $。实际上，如果$ n = 4 $和$ \ mathrm {char}（\ mathbb {f} _q）> 3 $，我们证明，$ x $中包含的任何理性线的理性点数均由$ o（| p | p | p |^{3/2+\ varepsilon}）$。从结果$ 6 $变量中，我们推断出$ n \ geq 7 $ over $ \ mathbb {f} _q（t）$时，$ n \ geq 7 $时，我们推断出微弱的近似值。 $ \ mathbb {f} _q（t）$当$ \ mathrm {char}（\ mathbb {f} _q）\ neq 3 $。我们的结果回答了关于有界高度的解决方案的数量到$ x_1^3+x_2^3+x_3^3 = x_4^3+x_5^3+x_6^3 $带有$ x_i \ in \ in \ mathbb {f} _q [t] $。
Given a non-singular diagonal cubic hypersurface $X\subset\mathbb{P}^{n-1}$ over $\mathbb{F}_q(t)$ with $\mathrm{char} (\mathbb{F}_q)\neq 3$, we show that the number of rational points of height at most $|P|$ is $O(|P|^{3+\varepsilon})$ for $n=6$ and $O(\lvert P \rvert^{2+\varepsilon})$ for $n=4$. In fact, if $n=4$ and $\mathrm{char}(\mathbb{F}_q) >3$ we prove that the number of rational points away from any rational line contained in $X$ is bounded by $O(|P|^{3/2+\varepsilon})$. From the result in $6$ variables we deduce weak approximation for diagonal cubic hypersurfaces for $n\geq 7$ over $\mathbb{F}_q(t)$ when $\mathrm{char}(\mathbb{F}_q)>3$ and handle Waring's problem for cubes in $7$ variables over $\mathbb{F}_q(t)$ when $\mathrm{char}(\mathbb{F}_q)\neq 3$. Our results answer a question of Davenport regarding the number of solutions of bounded height to $x_1^3+x_2^3+x_3^3 = x_4^3+x_5^3+x_6^3$ with $x_i \in \mathbb{F}_q[t]$.
