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

TL;DR

This paper establishes bounds on the number of rational points on diagonal cubic hypersurfaces over function fields, advancing understanding of rational solutions and weak approximation in this setting.

Contribution

It provides new bounds on rational points for diagonal cubic hypersurfaces over $F_q(t)$, including results on weak approximation and Waring's problem, answering a question of Davenport.

Findings

Bounded the number of rational points for $n=6$ and $n=4$.

Proved weak approximation for $n extgreater=7$.

Addressed Davenport's question on solutions to a specific cubic equation.

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