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), 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 (Gan Gross) to include all cubic rings (a cubic ring is a ring that is isomorphic to Z3 as a Z-module),[1] 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 [math]\displaystyle{ a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3 }[/math] 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.[2]
Examples
- Cubic plane curve
- Elliptic curve
- Fermat cubic
- Cubic 3-fold
- Koras–Russell cubic threefold
- Klein cubic threefold
- Segre cubic
Notes
- ↑ In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary scheme.
- ↑ Porteous, Ian R. (2001), Geometric Differentiation, For the Intelligence of Curves and Surfaces (2nd ed.), Cambridge University Press, pp. 350, ISBN 978-0-521-00264-6
References
- Delone, Boris; Faddeev, Dmitriĭ (1964), The theory of irrationalities of the third degree, Translations of Mathematical Monographs, 10, American Mathematical Society
- 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
- Hazewinkel, Michiel, ed. (2001), "Cubic form", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=c/c027260
- Hazewinkel, Michiel, ed. (2001), "Cubic hypersurface", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=c/c027270
- Manin, Yuri Ivanovich (1986), Cubic forms, North-Holland Mathematical Library, 4 (2nd ed.), Amsterdam: North-Holland, ISBN 978-0-444-87823-6, https://books.google.com/books?id=W03vAAAAMAAJ
 |  |
