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

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

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

• Delone, Boris; Faddeev, Dmitriĭ (1964) [1940, Translated from the Russian by Emma Lehmer and Sue Ann Walker], The theory of irrationalities of the third degree, Translations of Mathematical Monographs, 10, American Mathematical Society, MR0160744 
• Gan, Wee-Teck; Gross, Benedict; Savin, Gordan (2002), "Fourier coefficients of modular forms on G₂", Duke Mathematical Journal 115 (1): 105–169, doi:10.1215/S0012-7094-02-11514-2, MR1932327 
• Iskovskikh, V.A.; Popov, V.L. (2001), "Cubic form", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/c/c027260.htm 
• Iskovskikh, V.A.; Popov, V.L. (2001), "Cubic hypersurface", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/c/c027270.htm 
• Manin, Yuri Ivanovich (1986) [1972], Cubic forms, North-Holland Mathematical Library, 4 (2nd ed.), Amsterdam: North-Holland, ISBN 978-0-444-87823-6, MR833513, http://books.google.com/books?id=W03vAAAAMAAJ 

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