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Artin-Davenport Conjecture

The Artin-Davenport conjecture asserts that an integral cubic form in at least 10 variables has a nonzero integer zero. Thus, for a homogeneous polynomial F in Z[x_1,...,x_n] of degree 3 with n>=10, it asserts the existence of x in Z^n-{0} with F(x)=0.

Bonolis et al. (2026) reported a proof, combining geometric arguments with the circle method. The threshold is sharp: there are integral cubic forms in nine variables with no nonzero integer zero (Bonolis et al. 2026).

Bonolis et al. (2026) state that ChatGPT supplied a key incidence-dimension bound and assisted with a Galois-descent argument. Their accompanying Lean development checks the proof subject to five inputs from the literature; it is not a wholly self-contained formal proof.

See also

Cubic Form, Diophantine Equation, Homogeneous Polynomial

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References

Bonolis, D.; Browning, T.; Glas, J.; and Wang, V. Y. "The Artin-Davenport Conjecture on Cubic Forms." 6 Oct 2026. https://arxiv.org/abs/2610.08226.

Cite this as:

Weisstein, Eric W. "Artin-Davenport Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Artin-DavenportConjecture.html

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