Cubic form

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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 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]

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.
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References

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