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arXiv · 2610.08226

The Artin-Davenport conjecture on cubic forms

Dante Bonolis·Tim Browning·Jakob Glas·Victor Y. Wang
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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.

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Dante Bonolis, Tim Browning, Jakob Glas, Victor Y. Wang. 2026-10-06. The Artin-Davenport conjecture on cubic forms. https://arxiv.org/abs/2610.08226

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