In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 ,
Boris Delone
Boris Nikolayevich Delaunay or Delone (russian: Бори́с Никола́евич Делоне́; 15 March 1890 – 17 July 1980) was a Soviet and Russian mathematician, mountain climber, and the father of physicist, Nikolai Borisovich Delone. ...
and
Dmitry Faddeev
Dmitry Konstantinovich Faddeev ( rus, Дми́трий Константи́нович Фадде́ев, , ˈdmʲitrʲɪj kənstɐnʲˈtʲinəvʲɪtɕ fɐˈdʲe(j)ɪf; 30 June 1907 – 20 October 1989) was a Soviet mathematician.
Biography
Dmitry w ...
showed that binary cubic forms with integer coefficients can be used to parametrize
orders
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
in
cubic fields. Their work was generalized in to include all cubic rings (a is a
ring that is isomorphic to Z
3 as a
Z-module),
[In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary ]scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
. giving a
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
-preserving
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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
is linked to the classification of
umbilical points of surfaces. The
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of such cubics form a three-dimensional
real projective space and the subset of
parabolic form
Parabolic usually refers to something in a shape of a parabola, but may also refer to a parable.
Parabolic may refer to:
*In mathematics:
**In elementary mathematics, especially elementary geometry:
** Parabolic coordinates
**Parabolic cylindrica ...
s define a surface – the
umbilic torus.
Examples
*
Cubic plane curve
*
Elliptic curve
*
Fermat cubic
*
Cubic 3-fold
*
Koras–Russell cubic threefold
*
Klein cubic threefold
*
Segre cubic
Notes
References
*
*
*
*
*
*
Multilinear algebra
Algebraic geometry
Algebraic varieties
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