On a question of Davenport and diagonal cubic forms over $\mathbb{F}_q(t)$

Abstract

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 $x_i \in \mathbb{F}_q[t]$.