The singular series of a cubic form in many variables and a new proof of Davenport's Shrinking Lemma

verfasst von

Christian Bernert

Abstract

We study the singular series associated to a cubic form with integer coefficients. If the number of variables is at least 10, we prove the absolute convergence (and hence positivity) under the assumption of Davenport's Geometric Condition, improving on a result of Heath-Brown. For the case of nine variables, we give a conditional treatment. We also provide a new short and elementary proof of Davenport's Shrinking Lemma that has been a crucial tool in previous literature on this and related problems.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Artikel

Journal

Bulletin of the London Mathematical Society

Band

57

Seiten

681-691

Anzahl der Seiten

11

ISSN

0024-6093

Publikationsdatum

10.03.2025

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Allgemeine Mathematik

Fachgebiet (basierend auf ÖFOS 2012)

Zahlentheorie

Elektronische Version(en)

https://doi.org/10.1112/blms.13221 (Zugang: Offen)
https://doi.org/10.48550/arXiv.2310.02036 (Zugang: Offen)