The singular series of a cubic form in many variables and a new 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 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.