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 (Delone & Faddeev 1964), 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 (Gan, Gross & Savin 2002, §4) 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.

Examples

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.

References