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

TL;DR

This paper investigates the singular series of cubic forms with many variables, proving convergence and positivity under certain conditions, and introduces a new elementary proof of Davenport's Shrinking Lemma.

Contribution

It establishes absolute convergence of the singular series for cubic forms with at least 10 variables under Davenport's condition and offers a novel proof of Davenport's Shrinking Lemma.

Findings

Proves convergence and positivity of the singular series for ≥10 variables.

Provides a conditional result for 9-variable case.

Introduces a new elementary proof of Davenport's Shrinking Lemma.

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 $9$ variables, we give a conditional treatment. We also provide a new short and elementary proof of Davenport's Shrinking Lemma which has been a crucial tool in previous literature on this and related problems.