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

Given a non-singular diagonal cubic hypersurface $X\subset {\mathbb{P}}^{n-1}$ over ${\mathbb{F}}_q(t)$ with $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}({\mathbb{F}}_q)\mathrm{\ne }3$, we show that the number of rational points of height at most $\mathrm{\mid }P\mathrm{\mid }$ is $O(\mathrm{\mid }P{\mathrm{\mid }}^{3+\epsilon })$ for $n=6$ and $O(\mid P{\mid }^{2+\epsilon })$ for $n=4$. In fact, if $n=4$ and $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}({\mathbb{F}}_q)>3$ we prove that the number of rational points away from any rational line contained in $X$ is bounded by $O(\mathrm{\mid }P{\mathrm{\mid }}^{3\mathrm{/}2+\epsilon })$. From the result in $6$ variables we deduce weak approximation for diagonal cubic hypersurfaces for $n\ge 7$ over ${\mathbb{F}}_q(t)$ when $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}({\mathbb{F}}_q)>3$ and handle Waring’s problem for cubes in $7$ variables over ${\mathbb{F}}_q(t)$ when $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}({\mathbb{F}}_q)\mathrm{\ne }3$. 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 $x_i\in {\mathbb{F}}_q[t]$.