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

Glas, Jakob; Hochfilzer, Leonhard

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

Glas J, Hochfilzer L. On a question of Davenport and diagonal cubic forms over Fq(t). arXiv, 10.48550/arXiv.2208.05422.

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https://doi.org/10.48550/arXiv.2208.05422 [Preprint]

DOI

10.48550/arXiv.2208.05422

Preprint | Submitted | English

Author

Glas, Jakob^ISTA; Hochfilzer, Leonhard

Corresponding author has ISTA affiliation

Department

Browning Group

Abstract

Given a non-singular diagonal cubic hypersurface X⊂Pn−1 over Fq(t) with char(Fq)≠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(Fq)>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 Fq(t) when char(Fq)>3 and handle Waring's problem for cubes in 7 variables over Fq(t) when char(Fq)≠3. Our results answer a question of Davenport regarding the number of solutions of bounded height to x31+x32+x33=x34+x35+x36 with xi∈Fq[t].

Publishing Year

2022

Date Published

2022-08-10

Journal Title

arXiv

IST-REx-ID

18293

Cite this

Glas J, Hochfilzer L. On a question of Davenport and diagonal cubic forms over Fq(t). arXiv. doi:10.48550/arXiv.2208.05422

Glas, J., & Hochfilzer, L. (n.d.). On a question of Davenport and diagonal cubic forms over Fq(t). arXiv. https://doi.org/10.48550/arXiv.2208.05422

Glas, Jakob, and Leonhard Hochfilzer. “On a Question of Davenport and Diagonal Cubic Forms over Fq(T).” ArXiv, n.d. https://doi.org/10.48550/arXiv.2208.05422.

J. Glas and L. Hochfilzer, “On a question of Davenport and diagonal cubic forms over Fq(t),” arXiv. .

Glas J, Hochfilzer L. On a question of Davenport and diagonal cubic forms over Fq(t). arXiv, 10.48550/arXiv.2208.05422.

Glas, Jakob, and Leonhard Hochfilzer. “On a Question of Davenport and Diagonal Cubic Forms over Fq(T).” ArXiv, doi:10.48550/arXiv.2208.05422.

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