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

Given a non-singular diagonal cubic hypersurface \(X\subset \mathbb {P}^{n-1}\) $X\subset {\mathbb{P}}^{n-1}$ over \(\mathbb {F}_q(t)\) ${\mathbb{F}}_q(t)$ with \({{\,\textrm{char}\,}}(\mathbb {F}_q)\ne 3\) $\text{char}({\mathbb{F}}_q)\ne 3$ , we show that the number of rational points of height at most |P| is \(O(|P|^{3+\varepsilon })\) $O(|P{|}^{3+\epsilon })$ for \(n=6\) $n=6$ and \(O(|P |^{2+\varepsilon })\) $O(|P{|}^{2+\epsilon })$ for \(n=4\) $n=4$ . In fact, if \(n=4\) $n=4$ and \({{\,\textrm{char}\,}}(\mathbb {F}_q) >3\) $\text{char}({\mathbb{F}}_q)>3$ we prove that the number of rational points away from any rational line contained in X is bounded by \(O(|P|^{3/2+\varepsilon })\) $O(|P{|}^{3/2+\epsilon })$ . From the result in 6 variables we deduce weak approximation for diagonal cubic hypersurfaces for \(n\ge 7\) $n\ge 7$ over \(\mathbb {F}_q(t)\) ${\mathbb{F}}_q(t)$ when \({{\,\textrm{char}\,}}(\mathbb {F}_q)>3\) $\text{char}({\mathbb{F}}_q)>3$ and handle Waring’s problem for cubes in 7 variables over \(\mathbb {F}_q(t)\) ${\mathbb{F}}_q(t)$ when \({{\,\textrm{char}\,}}(\mathbb {F}_q)\ne 3\) $\text{char}({\mathbb{F}}_q)\ne 3$ . Our results answer a question of Davenport regarding the number of solutions of bounded height to \(x_1^3+x_2^3+x_3^3 = x_4^3+x_5^3+x_6^3\) $x_1^3+x_2^3+x_3^3=x_4^3+x_5^3+x_6^3$ with \(x_i \in \mathbb {F}_q[t]\) $x_i\in {\mathbb{F}}_q[t]$ .