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Cubic Form

A cubic form is a homogeneous polynomial of degree 3. In n variables it can be written

F(x_1,...,x_n)=sum_(i,j,k=1)^na_(ijk)x_ix_jx_k.

An integral cubic form has integer coefficients. The Artin-Davenport conjecture concerns nonzero integer zeros of integral cubic forms; Bonolis et al. (2026) reported its proof for n>=10.

See also

Artin-Davenport Conjecture, 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. "Cubic Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CubicForm.html

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