PreprintCC BY 4.0arXiv
The Artin-Davenport conjecture on cubic forms
Abstract
We show that any integral cubic form in ten or more variables has a nontrivial integer zero. Like earlier authors on this topic, we use geometric methods to produce a zero in some cases, and the circle method in all other cases. New ingredients are required on both fronts: in the first case, bounds on the dimension of singular loci and incidence varieties, and in the second case, a stratified geometric sieve, together with stratification results for point counts over finite fields and a cube-root Hensel lifting analysis.
Related
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BibTeX
@misc{bonolis2026artindavenport,
title = {{The Artin-Davenport conjecture on cubic forms}},
author = {Dante Bonolis and Tim Browning and Jakob Glas and Victor Y. Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08226},
eprint = {2610.08226},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08226},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Artin-Davenport conjecture on cubic forms
AU - Dante Bonolis
AU - Tim Browning
AU - Jakob Glas
AU - Victor Y. Wang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.08226
UR - https://arxiv.org/abs/2610.08226
AB - We show that any integral cubic form in ten or more variables has a nontrivial integer zero. Like earlier authors on this topic, we use geometric methods to produce a zero in some cases, and the circle method in all other cases. New ingredients are required on both fronts: in the first case, bounds on the dimension of singular loci and incidence varieties, and in the second case, a stratified geometric sieve, together with stratification results for point counts over finite fields and a cube-root Hensel lifting analysis.
ER -