Homogeneous polynomial of degree 3
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 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 & Savin 2002 , §4) to include all cubic rings (a cubic ring ring that is isomorphic to Z 3 as a Z -module[ 1] 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
{\displaystyle ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}}
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
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, p. 350, ISBN 978-0-521-00264-6
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, vol. 10, American Mathematical Society, MR 0160744 Gan, Wee-Teck; Gross, Benedict ; Savin, Gordan (2002), "Fourier coefficients of modular forms on G 2 ", Duke Mathematical Journal , 115 (1): 105–169, CiteSeerX 10.1.1.207.3266 doi :10.1215/S0012-7094-02-11514-2 , MR 1932327 Iskovskikh, V.A.; Popov, V.L. (2001) [1994], "Cubic form" , Encyclopedia of Mathematics EMS Press Iskovskikh, V.A.; Popov, V.L. (2001) [1994], "Cubic hypersurface" , Encyclopedia of Mathematics EMS Press Manin, Yuri Ivanovich (1986) [1972], Cubic forms ISBN 978-0-444-87823-6 MR 0833513
