Cubic form

Template:Short description 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 Template:Harv, 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 Template:Harv to include all cubic rings (a Template:Vanchor is a ring that is isomorphic to Z³ as a Z-module), 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 $a{x}^3+3b{x}^{2}y+3cx{y}^2+d{y}^3$ 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

References

• Template:Citation
• Template:Citation
• Template:Eom
• Template:Eom
• Template:Citation

Template:Algebraic-geometry-stub