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

In (Delone & Faddeev 1964), Boris Delone and DmitriÄ­ 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.

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 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.
  3. ↑ Porteous, Ian R. (2001), Geometric Differentiation, For the Intelligence of Curves and Surfaces (2nd ed.), Cambridge University Press, p. 350, ISBN 978-0-521-00264-6

References


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