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

verfasst von

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].

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Externe Organisation(en)

Pennsylvania State University

Typ

Artikel

Journal

Mathematische Annalen

Band

391

Seiten

5485–5533

Anzahl der Seiten

49

ISSN

0025-5831

Publikationsdatum

04.2025

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Allgemeine Mathematik

Elektronische Version(en)

https://doi.org/10.1007/s00208-024-03035-z (Zugang: Offen)