For every integer n≥ 10 and every homogeneous cubic form C∈Z[x_1,…,x_n], there exists a nonzero vector mathbfx∈Z^n such that C(mathbfx)=0.

Artin–Davenport conjecture on cubic forms

The Artin–Davenport program concerns nontrivial integer solutions of cubic equations, with the difficult target being uniform solubility in ten variables. The latest claim is by Dante Bonolis, Tim Browning, Jakob Glas, and Victor Y. Wang.

Known results

Davenport proved nontrivial integer solubility for all cubic forms with $n \ge 16$ (year not specified in the retrieved source).
Heath-Brown improved this to $n \ge 14$ (year not specified in the retrieved source).
For $n \ge 10$ , a 2023 paper proved absolute convergence and positivity of the singular series under Davenport’s geometric condition, but not uniform solubility.

October 2026 ten-variable claim

Bonolis, Browning, Glas, and Wang report that geometric methods, a stratified geometric sieve, finite-field point counts, and cube-root Hensel lifting establish the ten-variable cubic zero threshold. This is a claimed advance, not an independently verified resolution, and it does not settle every Hasse-principle formulation in the background.

Current status (as of October 2026): The bounds $n \ge 16$ and $n \ge 14$ are established, while the ten-variable result is claimed in a new preprint but remains unverified and the broader conjectural formulations remain open.

Artin–Davenport conjecture on cubic forms

Primary source

Additional references

Progress summary

Solutions 0