Cubic form explained

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 the case of a cubic form in three variables, the zero set is a cubic plane curve.

In, Boris Delone and Dmitry Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic fields. Their work was generalized in to include all cubic rings (a is a ring that is isomorphic to Z³ as a Z-module),^[1] 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

a x ^{3} + 3 b x ^{2} y + 3 c x y ^{2} + d y ^{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.

Examples

• Cubic plane curve
• Elliptic curve
• Fermat cubic
• Cubic 3-fold
• Koras–Russell cubic threefold
• Klein cubic threefold
• Segre cubic

Notes and References

1. In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary scheme.