# Cubic form

In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form.

In Template:Harv, Boris Delone and Dmitriĭ Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic fields. Their work was generalized in Template:Harv to include all cubic rings,[1][2] giving a discriminant-preserving bijection between orbits of a GL(2, Z)-action on the space of integral binary cubic forms and cubic rings up to isomorphism.

The classification of real cubic forms ${\displaystyle ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}}$ is linked to the classification of umbilical points of surfaces. The equivalence classes of such cubics form a three dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus or umbilic bracelet.[3]

## Notes

1. A cubic ring is a ring that is isomorphic to Z3 as a Z-module.
2. In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary scheme.
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## References

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