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

authored by

Jakob Glas, Leonhard Hochfilzer

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 x13+x23+x33=x43+x53+x63 with xi∈Fq[t].

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Pennsylvania State University

Type

Article

Journal

Mathematische Annalen

Volume

391

Pages

5485–5533

No. of pages

49

ISSN

0025-5831

Publication date

04.2025

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

General Mathematics

Electronic version(s)

https://doi.org/10.1007/s00208-024-03035-z (Access: Open)