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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 is a
ring that is isomorphic to Z 3 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
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}}
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
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 , North-Holland Mathematical Library, vol. 4 (2nd ed.), Amsterdam: North-Holland,
ISBN
978-0-444-87823-6 ,
MR
0833513