Details
| Original language | English |
|---|---|
| Pages (from-to) | 681-691 |
| Number of pages | 11 |
| Journal | Bulletin of the London Mathematical Society |
| Volume | 57 |
| Issue number | 3 |
| Publication status | Published - 10 Mar 2025 |
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.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
Research Area (based on ÖFOS 2012)
- NATURAL SCIENCES
- Mathematics
- Mathematics
- Number theory
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In: Bulletin of the London Mathematical Society, Vol. 57, No. 3, 10.03.2025, p. 681-691.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The singular series of a cubic form in many variables and a new proof of Davenport's Shrinking Lemma
AU - Bernert, Christian
PY - 2025/3/10
Y1 - 2025/3/10
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=86000425476&partnerID=8YFLogxK
U2 - 10.1112/blms.13221
DO - 10.1112/blms.13221
M3 - Article
VL - 57
SP - 681
EP - 691
JO - Bulletin of the London Mathematical Society
JF - Bulletin of the London Mathematical Society
SN - 0024-6093
IS - 3
ER -