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.

arXiv comment66 pages; improved exposition; added more discussion of Lemma 3.6 and connections to GIT; expanded overview of the paper in the introduction; streamlining of certain proofs