On a question of Davenport and diagonal cubic forms over \mathbbF_q(t)

2022/08/10 by Jakob Glas, Glas, Jakob, Leonhard Hochfilzer +1
Mathematics · #11D45 #11P05 #11P55 #11T55 #14G05 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2208.05422

openalex publication_date 2022/08/10 · openalex created_date 2022/08/12 · openalex updated_date 2026/07/28

Abstract

Given a non-singular diagonal cubic hypersurface X⊂ℙ^n-1 over \mathbbF_q(t) with char (\mathbbF_q)≠ 3, we show that the number of rational points of height at most |P| is O(|P|^3+ε) for n=6 and O(| P |^2+ε) for n=4. In fact, if n=4 and char(\mathbbF_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+ε). From the result in 6 variables we deduce weak approximation for diagonal cubic hypersurfaces for n≥ 7 over \mathbbF_q(t) when char(\mathbbF_q)>3 and handle Waring's problem for cubes in 7 variables over \mathbbF_q(t) when char(\mathbbF_q)≠ 3. Our results answer a question of Davenport regarding the number of solutions of bounded height to x₁³+x₂³+x₃³ = x₄³+x₅³+x₆³ with x_i ∈ \mathbbF_q[t].

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