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

Christian Bernert

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arXiv · 2310.02036

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

Christian Bernert

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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.

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Christian Bernert. 2023-10-03. The singular series of a cubic form in many variables and a new proof of Davenport's Shrinking Lemma. https://arxiv.org/abs/2310.02036

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