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

authored by

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Bulletin of the London Mathematical Society

Volume

57

Pages

681-691

No. of pages

11

ISSN

0024-6093

Publication date

10.03.2025

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

General Mathematics

Research Area (based on ÖFOS 2012)

Number theory

Electronic version(s)

https://doi.org/10.1112/blms.13221 (Access: Open)
https://doi.org/10.48550/arXiv.2310.02036 (Access: Open)